IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Weakly Relative Quasi-Injective Modules L. S Mahmood, A. S. Mijbass, K. S. Kalaf Departme nt of Mathematics, Ibn Al-Haitham College of Education, Unive rsity of Baghdad. Departme nt of Mathematics, College of Computer Science and Mathematics, Tik rit Unive rsity. Departme nt of Physics, College of Science, Unive rsity of Al-Anbar. Abstract: Let R be a commutative ring with unity and let M , N be unitary R-modules. In this research, we give generalizations for the concepts: weakly relative injectivity , relative tightness and weakly injectivity of modules. We call M weakly N-quasi-injective, if for each f  Hom(N,  ) there exists a submodule X of  such that f (N)  X ≈ M , where  is the quasi-injective hull of M . And we call M N-quasi-tight, if every quotient N / K of N which embeds in  embeds in M . While we call M weakly quasi-injective if M is weakly N-quasi- injective for every finitely generated R-module N. M oreover, we generalize some p rop erties of weakly N-injective, N-t ight and weakly injective modules to weakly N-quasi-injective, N-quasi-tight and weakly quasi-injective modules resp ectively. The relations among these concep ts are also st udied. Introduction The concept of weak relative injectivity of modules was introduced originally in [1]. Since then, the st udy of this concept has been illustrated extensively. We introduced in this research the concept of weak relative quasi-injectivity of modules as a gener alization of the concep t of weak relative injectivity which motivates our p rinciple subject of this research. This p aper contains five sections. In the first section, we introduced the concept of weakly relative quasi-injectivity of modules, where we call an R-module M weakly N- quasi- injective (N is any R-module) if for each f  Hom(N,  ) imp lies that f (N) is contained in T his paper represents a part of P h.D thesis written by the third author under the supervision of the first and the second authors and was submitted to the college of education Ibn-Al-Haitham university of Baghd ad. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 some submodule of  which is isomorphic to M , see definition 1.1. We established som e p rop erties of such modules. We showed that the class of such modules is not closed und er direct summand, see Ex. 1.6. While we could not p rove or disp rove that this class of modu les is closed und er dir ect sum. But we p roved a sp ecial case of this, see prop osition 1.7. Next we p roved that this class of modules is closed under essential extension, see p rop osition 1.15. The second section is devoted to give some characterizations of weakly relative qu asi- injective modules which are v ery useful in the n ext sections, see theorem 2.1, theorem 2.2, theorem 2.3, theorem 2.8, theorem 2.9, and theorem 2.10. In section three, we generalized the concept of relative tightness of modules which appeared in [2] into t he concep t of relative quasi-ti ghtness of modules, where we called an R- module M to be N-quasi – tight (N is any R-module) if and only if every quotient N / K of N which embeds in  embeds in M , see definition 3.2, we related this concept with the concept of relative quasi-injectivity of modules. It truns out that relative quasi-lightness of modules is a necessary condition for relative quasi- injectivity of modules, see prop osition 3.4, while the two concepts are equivalent in the class of uniform modules, see corollary 3.7 and corollary 3.8. We established in se ction four certain relations between quasi-ti ght modules and comp ressible modules in order to relate we ak relative quasi-in jectivity and co mpressibility of modules, where an R-module M is called compressible, if for every essential submodule N of M , M embeds in N, see [3]. Some of the results of this section were given in: Theorem 4.2, Corollary 4.3, Corollary 4.4 and corollary 4.5. In the last section of this p aper, we considered those modules which are we akly quasi- injective relative to each finitely generated module we would refer to any such module as bein g weakly -injective module. We would establish that: 1. An R-module M is weakly quasi-injective; i. If and only if M is weakly R n –quasi – injective for all p ositive integer n, se e theorem 5.3. ii. If and only if for all x1, x2, , xn   , there exists a submodule X of  such that xi  X ≈ M for all i = 1, 2, , n, see coro llary 5.5. 2. A ring R is weakly R n –quasi – injective if and on ly if for all x1, x2, , xn R , there exist s an element b  R such that annR(b) = 0 and xi  R b for all i = 1, 2, , n, see p rop osition 5.6. 3. A cyclic R-module is weakly quasi-injective if and on ly if it is weakly R 2 -quasi- injective, se e prop osition 5.8. Section One: Weakly Relative Quasi-Injective M odules We shall introduce in this section the concep t of weakly relative qusi- injectivity of modules. The relation between weakly relative qu asi-injective modules and certain typ es of modules are st udied. So me p rop erties of weakly relative quasi-injective modules are established. 1.1 De fini tion Let M and N be two R-modules. M is called weakly N-quasi-injective, if for each f  Hom(N,  ), there exists a submodule X of  such that f (N)  X ≈ M , where  is the quasi-injective hull of M . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 1.2 Remark Let M and N be two R-modules. Then i. If M is weakly N-injective, then M is weakly N-quasi-injective and the converse is not true in general. ii. If M is N-quasi-injective, then M is weakly N-quasi-injective and the converse is not true in general. iii. If M is quasi-injective, then M is weakly N-quasi-injective and the converse is not true in general. To disp rove the validity of the converse of the above remarks consider the following examp les resp ectively: 1.3 Example i. Let M = Z 2, N = Z and R = Z. Since Z2 is a quasi-injective Z-module, then Z2 is Z-quasi- injective and hence weakly Z-quasi-injective. However, Z2 is not weakly Z-injective, for if, f : Z  2  = E(Z2) (the injective hull of Z2), is such that f (a) = 3 2 a + Z for all a  Z, then f  Hom(Z, 2  ) and f (Z) ≈ Z 8 which is not embed in Z2. ii. Let M =Z, N = 2Z, R = Z and f : 2Z  Q is such that f (2a) = 2 5 a + Z for all a  Z, then f  Hom(2Z, Q). We take X = 2 ( ) 5 the submodule of Q generated by 2 5 and consequently f (2Z)  2 ( ) 5 . Hence Z is weakly 2Z-quasi-injective. However, Z is not 2Z-quasi-injective, since f (2Z)  Z. iii. Let M = Z, N = 2Z and R = Z. Then Z is weakly 2Z-quasi-injective, but Z is not quasi- injective. 1.4 Proposi tion Let M and N be two R-modules and let I be an ideal of R such that I  annR(  )  annR(N). T hen M is weakly N-quasi-injective R-module if and only if M is weakly N-quasi- injective R / I-module. Proof: I  annR(  )  annR(N), imp lies that M and N are R / I-modules. M oreover, f :N   is an R-homomorphism if and only if f is an R / I –homomorp hism, and X is an R- submodule of  if and only if X is an R / I – submodule of  . Hence the details of the p roof are followed directly by using the definition 1.1. 1.5 Remark A direct summand of weakly relative quasi-injective module is not weakly relative quasi- injective in general, as it is shown in the following examp le. 1.6 Example Let M =Z  Q, N = Q and R = Z. Let f  Hom(Q, Q ) = Hom(Q,Q  Q). If f = 0, the proof is obvious. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 If f  0, then f is a monomorp hism, for if x  Q and f (x) = 0 with a x b  and a, b  Z, a  0, b  0, then 0 = f ( ) a b = a f 1 ( ) b imp lies that f 1 ( ) b = 0. Now, f (1) = f ( ) b b = b f 1 ( ) b = 0. Hence f (Q) = 0, which is a contradiction. So f is a monomorp hism. Therefore f (Q) = 0  A or f (Q) = B  0 or f (Q)={(x,x): xQ}, where A and B are submodules of Q. If f (Q) = 0  A, t hen f (Q)  0  Q  Z  Q  Q  Q. T ake X = M = Z  Q, t hen f (Q)  X ≈ M = Z  Q. Similarly , if f (Q) = B  0. If f (Q)={(x,x): xQ} ≈ Q, take Y = {(x,x): xQ}  Q  Q. It is easy to p rove that Z  Y ≈ Z  Q. T herefore f (Q)  Y  Z  Y ≈ Z  Q. Hence Z  Q is weakly Q-quasi-injective. But it is clear that Z is not weakly Q-quasi-injective. 1.6 Remark We can not p rove and we can not disp rove that the class of weakly relative quasi- injective modules is closed under direct sum. However, we give a sp ecial case of this. 1.7 Proposi tion Let M and N be two R-modules, such that L M = L M . If L and M are weakly N- quasi-injective, then L  M is also weakly N-quasi-injective. Proof: Let f  Hom(N, L M ). Then f  Hom(N, L M ). But Hom(N, L M ) ≈ Hom(N, L )  Hom(N, M ) by [4]. Hence f = (,) with  Hom(N, L ) and  Hom(N, M ). Therefore there exists submodules X and Y of L and M resp ectively, such that (N)  X ≈ L and (N)  Y ≈ M . On t he other hand, (N) ≈ (N)  0  X  Y ≈ L  M , and (N) ≈ 0  (N)  X  Y ≈ L  M . Now, f (N) = (,)(N) = ((N),(N))  X  Y ≈ L  M , which comp letes the proof. 1.8 Remark If L, M and N are R-modules, such that M is weakly N-quasi-injective and M is weakly L-quasi-injective, then it is not true in general that: i. M is weakly N  L - quasi-injective. ii. M is weakly N + L - quasi-injective. Consider the following examp les: i. Let M = L = N = Z and R = Z. Then Z as a Z-module is weakly - quasi - injective. But Z is not weakly Z  Z - quasi- injective. In fact, if we define f : Z  Z  Q by f (a,b) = 2 a + 3 b where a, b  Z, then it can be easily seen that f  Hom(Z  Z,Q) and f (Z  Z) = (( 1 2 , 1 3 )) ≈ Z . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 ii. Let M = Z, N = ( 1 2 ), L = ( 1 3 ) and R = Z. Then Z as a Z-module is weakly ( 1 2 )-quasi- injective and Z is weakly ( 1 3 )-quasi-injective. But Z is not weakly ( 1 2 ) + ( 1 3 )-quasi- injective. For if, we define f : ( 1 2 ) + ( 1 3 )  Q by f ( 2 a + 3 b ) = 2 a + 3 b where a, b  Z, then it can be easily shown that f  Hom(( 1 2 ) + ( 1 3 ),Q) and f (( 1 2 ) + ( 1 3 )) = (( 1 2 , 1 3 )) ≈ Q. 1.9 Proposi tion Let M be an R-module and N be a submodule of M . If M is weakly N-quasi-injective, then M is weakly L-quasi-injective for each submodule L of N. Proof: Let f  Hom(L, M ). Consider the following diagram: L i j      f      M where i and j are the inclusion homomorp hisms and the homomorp hism  which makes the diagram commutative exists because M is quasi-injective. Therefore  i j = f . let  = N : N  M . So there exists a submodule X of M such that (N)  X ≈ M . But f (L)  (N), t hus f (L)  X ≈ M and hence M is weakly L-quasi-injective. 1.10 Corollary Let L and N be two submodules of an R-module M such that L  N. If M is weakly N- quasi –injective, then M is weakly L-quasi-injective. 1.11 Corollary Let M be an R-module and N be a submodule of M . If L is a submodule of M and M is weakly N-quasi-injective, then M is weakly N  L - quasi-injective. In the following two results, we exp lain the behavior of weakly -quasi-injectivity under homomorp hism. 1.12 Proposi tion Let H, N and M be R-modules and let g : N  H be an epimorp hism. If M is weakly N-quasi-injective, then M is weakly H-quasi-injective. Proof: Let f  Hom(H, M ). Then f  g  Hom(N, M ). So there exists a submodule X of M such that f (g(N))  X ≈ M which means t hat M is weakly H-quasi-injective. 1.13 Corollary Let N be a submodule of an R-module M and let g: M  M be an ep imorp hism. If M is weakly N-quasi-injective, then M is weakly g(N)-quasi-injective. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Recall that, a submodule N of an R-module M is called quasi-invertible if Hom(M / N, M ) = 0, [5]. 1.14 Proposi tion Let M be a torsion-free R-module and let N be a quasi-invertible submodule of M . If M is weakly M / N-quasi-injective, then N is a quasi-invertible submodule of M . Proof: Assume that N is not quasi-invertible in M . Then Hom( M /N, M )  0. Let f : M / N  M be a non-zero homomorphism. Therefore there exists m + N  M / N with m  M and m  N such that 0  f (m + N) = x for some x  M . Let i : M / N  M / N be the inclusion homomorp hism. Then f  i  Hom(M / N, M ). So there exists a submodule X of M such that f  i (M / N )  X ≈ M . let g : X  M be an isomorphism, then g f i  Hom(M / N,M ) = 0. Therefore g f i = 0 imp lies that f i = 0 and hence f (M / N) = 0. But m  M and m  N and M is essential in M , so t here exists 0  r  R such that r m  M . Hence r m + N  M / N and f (r m + N) = 0 = r f (m) + N = r x imp lies that r = 0 which is a contradiction. Therefore N is quasi-invertible in M . 1.15 Proposi tion Let M and N be two R-modules and let L be an essential extension of M . If M is weakly N-quasi-injective, then L is also weakly N-quasi-injective. Proof: Let f  Hom(N, L ). But L = M [by cor.19.8, p.65, [6]]. Hence f  Hom(N, M ). So there exists a submodule X of M such that f (N)  X ≈ M . Consider the following diagram: 32 L L iig        i1                L where g : X  M be an isomorp hism and i1, i2, i3 are inclusion homomorphisims. L being quasi-injective, so there exists a homomorp hism  : L  L such that  i g = i1 with i = i3 i2. We claim that ker  = {0}. Let 0  ℓ  L and (ℓ) = 0. But M is essential in L , so t here exists 0  r  R such that 0  r ℓ  M . Hence there exists x  X such that g(x) = r ℓ. Now, x =  i g (x) =  (r ℓ) = r (ℓ) = 0. So, r ℓ = 0 which is a contradiction. Therefore  is a monomorphism. Let  = L. Then  i g = i1. Hence X  (L) ≈ L. Therefore f (N)  X  (L), which means that L is weakly N-quasi-injective. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Section Two: Characterizations of Weak ly Relative Quasi Injective Module s We give in this section many interesting characterizations of weakly relative quasi- injective modules which are very useful in the next sections. First , we shall show that t he concep t of weakly quasi-injectivity can be given in terms of commutative diagram. 2.1 The orem Let M and N be two R-modules. Then M is weakly N-quasi-injective if and only if every element f  Hom(N,  ) can be written as a comp osition g h where h : N  M is a homomorp hism and g : M   is a monomorp hism. That is the following diagram is commutative. hom o. h      f   g    Proof: Assume that M is weakly N-quasi-injective. Let f  Hom(N,  ). Then there exists a submodule X of  such that f (N)  X ≈ M . So, f : N  X is a homomorp hism. Let  : X  M be an isomorphism. We take h =  f. Then h : N  M is a homomorphism. Let g = i  – 1 where i : X   is the inclusion homomorp hism. Hence g : M   is a monomorp hism. Now, g  h = (i  – 1)  ( f.) = i f = f which proves the "only if” p art. To p rove the "if” p art: Let f  Hom(N,  ). By hy p othesis, there exists a homomorp hism h : N  M and a monomorp hism g : N   such that f = g h. We take X = g(M ). Then X is a submodule of  and X ≈ M , moreover, f (N) = g(h(N))  g(M ) = X ≈ M . Therefore M is weakly N- quasi-injective. The following concept is needed for our next result. Let M and N be two R-modules. M is called N-cyclic, if M is isomorphic to N / K for some submodule K of N, [7]. 2.2 The orem Let M and N be two R-modules. Then M is weakly N-quasi-injective if and only if for any N-cy clic submodule X of  there exists a submodule L of  such that X  L ≈ M . Proof: Assume that M is weakly N-quasi-injective. Let X be an N-cy clic submodule of  . So, X ≈ N / K for some submodule K of N. Then we have: / i        where  is the natural homomorp hism,  is an isomorphism and i is the inclusion homomorp hism. Let f = i   . Then f  Hom(N,  ), imp lies that there exists a homomorp hism h : N  M and a monomorp hism g : N   such that f = g h (by Theorem 2.1). IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Now, g h(N) = f (N) = i (N) = i(N / K) = i(X) = X. T herefore g h(N) = X. We take L = g(N) t o obtain that L is a submodule of  and L ≈ M . M oreover X = g(N)  g(M ) = L. Conversely , To p rove M is weakly N-quasi-injective. Let f  Hom(N,  ). Then f (N) is a submodule of  and f (N) ≈ N / ker f. That means f (N) is an N-cyclic submodule of  . Therefore there exists a submodule L of  such that f (N)  L ≈ M , and hence the result follows. 2.3 The orem Let M and N be two R-modules. Then the following st atements are equivalent: 1. M is weakly N-quasi-injective. 2. For any submodule K of N, M is weakly N / K-quasi-injective. 3. For any submodule K of N and any homomorp hism f : N / K   , there exists a monomorp hism g: M   and a homomorp hism h: N / K  M such that g h = f. Proof: (1)  (2) Let K be a submodule of N and let f  Hom(N / K,  ). Let : N  N / K be the natural homomorp hism. Then f   Hom(N,  ) and hence by (1), there exists a submodule X of  such that f (N)  X ≈ M . Therefore f (N / K)  X ≈ M which proves (2). (2)  (3) We follow as in the proof of theorem 2.1. (3)  (1) Let f  Hom(N,  ) and let K = ker f . Then define f : N / K   by f (a + K) = f (a) for all a  N. f is a homomorp hism. It can be easily shown that f is a monomorp hism. Hence by (3), there exists a monomorp hism g : M   and a homomorp hism h : N / K  M such that g h = f . Now, f (N) = f (N/K) = g(h(N/K))  g(M ). We take X = g(M ), imp lies that f (N)  X ≈ M , which proves (1). The following lemma is needed in order to give some ap p lications of theorem 2.3. 2.4 Lemma Let K, M and N be R-modules with N ≈ K. If M is weakly N-quasi-injective, then M is weakly K-quasi-injective. Proof: Is obvious, so it is omitt ed. 2.5 Corollary Let K, M and N be R-modules. If M is weakly K-quasi-injective and N is K-cy clic. Then M is weakly N-quasi-injective. Proof: M being K-quasi-injective, imp lies that M is weakly K / L-quasi-injective for every submodule L of K (by theorem 2.3). But N is K-cy clic, so N ≈ K / L for some submodule L of K. Hence M is weakly N-quasi-injective (by lemma 2.4). 2.6 Corollary If M is weakly N-quasi-injective R-module and A is a direct summand of N, then M is weakly A-quasi-injective. Proof: follows easily by using theorem 2.3 and lemma 2.4. As a consequence of 2.6 we have the following result: IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 2.7 Corollary Let M and N be two R-modules such that N is quasi-injective and M is weakly N-quasi- injective. Then M is weakly A-quasi-injective for every closed submodule A of N. Proof: N being quasi-injective and A is a closed submodule of N imp lies that A is a direct summand of N [see cor. 16.9, p.64, [6]]. Hence the result follows by 2.6. The following theorem characterizes weakly -quasi-injectivity relative to the R-module R. 2.8 The orem Let M be an R-module. Then M is weakly R-quasi-injective if and only if for each element x   , there exists a submodule X of  such that x  X ≈ M . Proof: Assume that M is weakly R-quasi-injective. Let x   . Define f : R   by f (r) = r x for each r  R. Clearly f is well-defined R-homomorp hism. Thus there exists a submodule X of  such that f (R)  X ≈ M . But x = 1x = f (1)  f (R). Hence x  X which is what we wanted. Conversely , Let f  Hom(R,  ). Then f (1)   . Let x = f (1). Hence there exists a submodule X of  such that x  X ≈ M . It is left to show that f (R)  X. Let a  f (R), then a = f (r) for some r  R. a = f (r) = r f (1) = r x  X. T herefore f (R)  X and hence M is weakly R-quasi-injective. As a sp ecial case, we shall characterize the weakly quasi-injectivity of the R-module R relative to itself. 2.9 The orem R is weakly R-quasi-injective R-module if and only if for each element a  R , there exists an element b  R such that a  R b and annR(b) = 0. Proof: Assume that R is weakly R-quasi-injective R-module. Let a  R . Define f : R  R by f (r) = r a for each r  R. It can be easily shown that f is a well-defined R- homomorp hism. Hence there exists a submodule X of R such that f (R)  X ≈ R. Clearly , f (R) = R a. Thus R a  X, implies that a = 1a  X. Let : R  X be an isomorp hism. So there exists an element c  R such that a = (c). Hence a = (c1) = c (1) = c b  R b where b = (1). Therefore a  R b. Now, let r  annR(b). Then r b = 0 and hence 0 = r (1) = (r) imp lies that r = 0. Hence annR(b) = 0. Conversely , Let f  Hom(R, R ). Then f (1)  R . Let f (1) = a. So there exists an element b  R such that a  R b and annR(b) = 0. We take X = R b imp lies that X  R . But R b ≈ R / annR(b) ≈ R. M oreover f (R) = {f (r) : r  R} = {r f (1) : r  R} = R a  R b. Therefore f (R)  X ≈ R. This comp letes the proof. We shall establish in the following theorem a general case of theorem 2.9. 2.10 The orem R be an integral domain. Let M and N be two cyclic torsion-free R-modules. Then M is weakly N-quasi-injective if and only if for each element x   there exists an element y   such that x  Ry and annR(y) = 0. Proof: Assume that M is weakly N-quasi-injective. Let x   . Sup p ose that M = (m) and N = (n) for some m  M and n  N. Define f : N   by f (r n) = r x for all r  R. f is well- IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 defined homomorphism. Therefore there exists a submodule X of  such that f (N)  X ≈ M . Let y = f (n). Then y = x   and x = 1y  R y. Now, if t  annR(y), then t y = 0. Let g : X  M be an isomorphism, implies that 0 = g(ty) = t g(y) and hence t = 0. Thus annR(y) = 0. Conversely , Let f  Hom(N,  ), and let x = f (n). Then x   and hence there exists an element y   such that x  R y and annR(y) = 0. Let X = (x). Then X is a submodule of  and f (N)  X. We claim that X ≈ M . Define h : M  X by h(r m) = r x for all r  R. It is clear that h is a well-defined homomorp hism. M oreover if r x = 0, imp lies that r (t y) = 0 for some t  R. Thus r t = 0. If r = 0, we have done. If t = 0, then x = 0 which is a contradiction. Thus h is a monomorp hism. Clearly h is an epimorp hism. Hence X ≈ M and therefore M is weakly N- quasi-injective. When we weaken the conditions in theorem 2.10, we get t he following result: 2.11 Proposi tion Let M and N be two cy clic R-modules. If M is torsion-free, then M is weakly N-quasi- injective. Proof: Let M = (m) and N = (n) for some m  M and n  N. Let f  Hom(N,  ) and let x = f (n). Then x   . Sup p ose that X = (x). Then X is a submodule of  and f (N)  X. Define g : X  M by g(r x) = r m for all r  R. If r x = 0, we claim that r = 0. We have x   and M is an essential submodule of  , so t here exists a non-zero element t  R such that t x  M . Hence annR(t x) = 0. But annR(x)  annR(t x), so annR(x) = 0. Hence r = 0, thus g is well-defined. It can be easily shown that g is an isomorphism. Therefore X ≈ M and hence the result follows. 2.12 Remark The converse of p rop osition 2.11, may not be true in general, consider the following examp le: 2.13 Example Let M = Z 4, N = Z and R = Z. Then M is weakly N-quasi-injective. But M is not torsion- free R-module. On t he other hand, examp le 1.6 shows t hat t he condition N is cyclic in prop osition 2.11, can not be dropp ed. Section Three: Weak ly Re lative Quasi-Injective Module s and Quasi- Tight Module s We introduce in this section the concept of relative quasi-tightness of modules and we st udy the relation of this concept with the concept of relative weakly quasi-injectivity of modules. 3.1 De fini tion Let M and N be two R-modules. We say that M is N-quasi-tight if and only if every quotient N / K of N which embeds in  embeds in M . IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 M is called R-quasi-tight if and only if for every ideal I of R, every quotient R / I of R which embeds in  embed in M . 3.2 De fini tion An R-module M is called quasi-tight if M is N-quasi-tight for every finitely generated R- module N. 3.3 Remark Every N-t ight R-module is N-quasi-tight and the converse is not true in general. Consider the following examp le: Let M = Z 2, N = Z, and R = Z. Let K be a submodule of N. If N / K embeds in 2 = Z 2, so M is N-quasi-tight. Now, let K = 4Z. Thus Z / 4Z ≈ Z4 embeds in 2  = E(Z2), but Z4 can not embeds in Z2. Whence M is not N-t ight. 3.4 Proposi tion Let M and N be two R-modules. If M is weakly N-quasi-injective, then M is N-quasi- tight. Proof: Let K be a submodule of N such that N / K embeds in . Then there exists a monomorp hism f : N / K   . Let  : N  N /K be the natural homomorp hism. Hence f   Hom(N,  ), so t here exists a submodule X of  such that (f )(N)  X ≈ M . Hence f (N / K)  X ≈ M which implies that f : N / K  X is a homomorp hism. Let g : X  M be an isomorphism. Then g f : N / K  M is a monomorp hism. Which comp letes the p roof. 3.5 Corollary Let M be an R-module. If M is weakly R-quasi-injective, then M is R-quasi-tight. Recall that, if A and B are submodules of an R-module C, such that A is a maximal submodule of C with the prop erty that A  B = 0, then A is called a complement of B in C, [8]. 3.6 The orem Let M and N be two R-modules. Then M is weakly N-quasi-injective if and only if for each submodule L of N and for every monomorp hism f : N / L   , we have: i. There exists a monomorp hism f : N / L  M , and ii. For every comp lement K of f (N / L) in M , there exists a submodule K of  such that K  f (N / L) = 0 and K ≈ K. Proof: Assume that M is weakly N-quasi-injective. Let L be a submodule of N and let f : N / L   be a monomorp hism, M being weakly N-quasi-injective imp lies that M is weakly N / L-quasi-injective (by theorem 2.3) and hence there exists a homomorp hism f : N / L  M and there exists a monomorp hism  : M   such that  f = f (by theorem 2.1). But f is a monomorphism, therefore f  is also a monomorphism. Thus (i) follow. To verify (ii), let K be a comp lement of f (N / L) in M . Let K = (K). Then K is a submodule of  . We claim that K  f (N / L) = 0. Let x  K  f (N / L) and x  0. Hence there exists 0  y  K such that x = (y) and there exists 0  z  N / L such that x = f (z). Therefore (y) = f (z) and hence (y) = ( f (z)), but  IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 is a monomorphism, so y = f (z), imp lies that 0  y  K  f (N / L) which is a contradiction. Hence K  f (N / L) = 0, so (ii) is also hold. Conversely , Let us assume that (i) and (ii) are hold. Let L be a submodule of N and let f : N / L   be a monomorp hism. By (i), there exists a monomorp hism f : N / L  M . Let K be a complement of f (N / L) in M . By (ii), there exists a submodule K of  such that K  f (N / L) = 0 and K ≈ K. Let h : K K be an isomorphism and define  : f (N / L)  K   by ( f (x) + k) = f (x) + h(k) for all x  N / L, for all k  K. T hen  is well-defined homomorphism, moreover, if f (x) + h(k) = 0 for some x  N / L and k  K, implies that f (x) = – h(k)  K  f (N / L) = 0. So, f (x) = 0 and h(k) = 0, hence  is a monomorp hism. Therefore  is extended to a monomorp hism : M   . We claim that  f = f. Let x  N / L. Then ( f (x)) = ( f (x)) = ( f (x) + 0) = f (x). Hence  f = f and so, M is weakly N-quasi-injective (by theorem 2.1). 3.7 Corollary Let M and N be two R-modules. If M is uniform and N-quasi-tight, then M is weakly N- quasi-injective. Proof: Let L be a submodule of N let f : N / L   be a monomorp hism. But M is N- quasi-tight, therefore there exists a monomorp hism f : N / L  M and hence (i) in theorem 3.6 holds. Now, if L = N, then f (N / L) = 0 and hence M is a comp lement of f (N / L) in M and M  f (N / L) = 0. If L  N, t hen f (N / L) is a non-zero submodule of M and since M is uniform imp lies that 0 is the only comp lement of f (N / L) in M , and hence (ii) in theorem 3.6 is also hold. Therefore M is weakly N-quasi-injective (by theorem 3.6). 3.8 Corollary Let M and N be two R-modules such that M is uniform. Then M is N-quasi-tight if and only if M is weakly N-quasi-injective. Proof: follows by p rop osition 3.4 and corollary 3.7. Section Four: Quasi-Tight Module s and Compressible Module s In this section, we establish some relations between relative quasi-tight modules and comp ressible modules in the class of quasi-injective modules. 4.1 De fini tion An R-module M is called comp ressible if for all non-zero submodules N of M , M embeds in N, [9]. In general, an R-module M is comp ressible if for every essential submodule N of M , M embeds in N, [4]. First , we establish the relationship between relative quasi-tight modules and comp ressible modules. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 4.2 The orem Let M be a quasi-injective R-module and let N be any R-module. Then every submodule of M is N-quasi-tight if and only if every quotient N / K of N which embeds in M is comp ressible. Proof: Assume that every submodule of M is N-quasi-tight. Let N / K be embeds in M . Hence there exists a monomorp hism f: N / K  M . We have to show that N / K is comp ressible. Let L be an essential submodule of N / K. It can be easily seen that f (L) is essential in f (N / K) and hence ( L) ( / L)f f  [by cor.19.8, p .65, [6]]. Since N / K embeds in ( / L) (L)f f  and f (L) is N-quasi-tight, we get that N / K embeds in f (L) ≈ L. Thus N / K embeds in L which is what we wanted. Conversely , Sup p ose that every quotient of N which embeds in M is comp ressible. Let A be a submodule of M . We have to show that A is N-quasi-tight. Let K be a submodule of N and let h : N / K   be a monomorp hism. But    = M , imp lies that i h : N / K  M is a monomorp hism where i :    is the inclusion homomorp hism. Let B = h(N / k)  A. Then B  0. We claim that B is essential in h(N / K). For if, B  C = 0 for some non-zero submodule C of h(N / K), then 0 = (h(N / K)  A)  C = A  C which is a contradiction. Therefore 0  B is essential in h(N / K), which implies that h – 1 (B) is essential in N / K. But N / K is comp ressible therefore N / K embeds in h – 1 (B). On t he other hand h – 1 (B) ≈ B A. T hus N / K embed in A, as desired. 4.3 Corollary Let M be a quasi-injective R-module. Then every submodule of M is quasi-tight if and only if every finitely generated submodule of M is comp ressible. Proof: Assume that every submodule of M is quasi-tight. Let A be a finitely generated submodule of M . Then A is N-quasi-tight for every finitely generated R-module N. Therefore M is A-quasi-tight and according to theorem 4.2. We get t hat for each submodule B of A such that A / B embeds in M is comp ressible. But A is finitely generated imp lies that A / B is also finitely generated. Hence every finitely generated submodule of M is comp ressible. Conversely , Assume that every finitely generated submodule of M is comp ressible. To p rove that every submodule of M is quasi-tight. Let A be a submodule of M and let N be a finitely generated R-module. Let K be a submodule of N such that N / K embeds in  . But N / K is a finitely generated R-module which embeds in M , so by hy p othesis, N / K is comp ressible. Therefore A is N-quasi-tight for each finitely generated R-module N (by theorem 4.2). Hence A is A- quasi-tight. 4.4 Corollary Let M be a quasi-injective R-module. Then every submodule of M is weakly R-quasi- injective if and only if every cyclic submodule of M is comp ressible. Proof: Assume that every submodule of M is weakly R-quasi-injective. Then every submodule of M is R-quasi-tight (by corollary 3.5) and according to theorem 4.2, we get that every quotient R / I of R (with I is an ideal of R) which embeds in M is comp ressible. Now, let A = (a) be a cyclic submodule of M for some a  M . Then A ≈ R / annR(a). So, R / annR(a) is comp ressible. Hence A is compressible. Conversely , Assume that every cyclic submodule of M is comp ressible. Because of the fact that every cyclic submodule of M can be writt en as a quotient R / I for some ideal I of R, and hence for each ideal I of R, if R / I embeds in M is comp ressible, therefore every cyclic submodule of M is R-quasi-tight (by theorem 4.2). To p rove every submodule of M is weakly R-quasi- IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 injective. Let A be a submodule of M and let x   . Then (x) ≈ R / annR(x)   . But every cyclic submodule of M is R-quasi-tight, hence (x)  A. We take X = A implies that x  X = A. T hus A is weakly R-quasi-injective (by theorem 2.8). 4.5 Corollary Let N be an R-module. If every R-module is N-quasi-tight, then N / K is comp ressible for every submodule K of N. Proof: Assume that every R-module is N-quasi-tight. Let K be a submodule of N and let A = /  . By hy p othesis, we get t hat every submodule of A is N-quasi-tight, and since A is a quasi-injective R-module, imp lies that N / K is comp ressible for every submodule K of N (by theorem 4.2). Section Five: Weak ly Quasi-Injective Module s In this section, we shall concentrate on considering those modules which are weakly quasi-injective relative to each finitely generated module; we shall refer to any such module as being weakly quasi-injective module. 5.1 De fini tion An R-module M is called weakly quasi-injective, if M is weakly N-quasi-injective for every finitely generated R-module N. Equivalently , M is weakly quasi-injective if and only if for each finitely generated R-module N and for each f  Hom(N,  ), there exists a submodule X of M such that f (N)  X ≈ M . 5.2 Remarks 1. A ring R is called weakly quasi-injective if and only if the R-module R is weakly quasi- injective. 2. Every weakly injective R-module is weakly quasi-injective and the converse is not true in general, see examp le 1.3. 5.3 The orem Let M be an R-module. Then M is weakly quasi-injective if and only if M is weakly R n - quasi-injective for all p ositive integer n. Proof: the "only if” p art is obvious. To p rove the "if” p art. Let N be a finitely generated R-module. We have to show that M is weakly N-quasi-injective. Sup p ose that N = Ra1 + Ra2 + + Ran where ai  N for all i = 1, 2, , n. Define f : R n  N such that f (r1, r2, , rn) = r1 a1 + r2 a2 +  + rn an for all r1, r2, , rn  R. It can be easily checked that f is well-defined epimorp hism. Therefore, R n / ker f N. But M is weakly R n -quasi-injective, imp lies that M is weakly R n / ker f-quasi-injective (by theorem 2.3). Therefore M is weakly N-quasi –injective. 5.4 Proposi tion An R-module M is weakly R n -quasi-injective if and only if for all x1, x2, , xn   , there exists a submodule X of  such that xi  X ≈ M for all i = 1,2, , n. Proof: Assume that M is weakly R n -quasi-injective. Let x1, x2, , xn   . Let N = Rx1 + Rx2 + + Rxn. Thus N is a finitely generated R-module. In fact N is a submodule of  . Let j: N   be the inclusion homomorp hism. By theorem 5.4, M is weakly N-quasi- injective, therefore there exists a submodule X of  such that j(N)  X ≈ M . Hence xi  X for all i = 1, 2, , n which completes the proof of the first p art. Conversely , IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 We have to show that M is weakly R n -quasi-injective. Let f  Hom(R n ,  ). Sup p ose that f (1,0,0,,0) = x1, f (0,1,0,,0) = x2, , f (0,0,0,,1) = xn. Then x1, x2, , xn   . So, by hy p othesis there exists a submodule X of  such that xi  X ≈ M for all i = 1, 2, , n which imp lies that f (R n )  X ≈M and hence M is weakly R n -quasi-injective. 5.5 Corollary An R-module M is weakly quasi-injective if and only if for all x1,x2,,xn , there exists a submodule X of  such that xi  X ≈ M for all i = 1, 2, , n. 5.6 Proposi tion A ring R is weakly R n -quasi-injective if and only if for all x1, x2, , xn  R there exists an element b  R such that annR(b) = 0 and xi  Rb for all i =1,2,,n. Proof: Sup p ose that R is weakly R n -quasi-injective. Let x1, x2, , xn  R . By p rop osition 5.4., there exists a submodule X of R such that xi  X ≈ R, for all i = 1, 2, , n. Let  : R  X be an isomorp hism. Put b = (1). Then b  R and for all i = 1, 2, , n, xi = (ri) for some ri  R and hence xi = ri (1) = ri b for all i = 1, 2, , n. Therefore xi  Rb for all i = 1, 2, , n. M oreover, if r b = 0 for some r  R, imp lies that r = 0 and hence annR(b) = 0. Conversely , We have to show that R is weakly R n -quasi-injective. Let f  Hom(R n , R ). Let f (1,0,0,,0) = x1, f (0,1,0,,0) = x2, , f (0,0,0,,1) = xn. Then x1, x2, , xn  R and hence there exists b  R such that xi  Rb for all i = 1, 2, , n and annR(b)=0. Let X = Rb. Then X is a submodule of R , xi  X for all i = 1, 2, , n and X ≈ R. Therefore R is weakly R n -quasi- injective (by p rop osition 5.4). The following corollary is also a consequence of theorem 5.3 and p rop osition 5.6. 5.7 Corollary A ring R is weakly quasi-injective if and only if for all x1, x2, , xn  R there exists an element b  R such that annR(b) = 0 and xi  Rb for all i = 1, 2, , n. Finally, we give the following characterization. 5.8 Proposi tion A cyclic R-module is weakly quasi-injective if and only if it is weakly R 2 -quasi- injective. Proof: the "only if " p art is obvious. To p rove the "if " p art, let M be a cyclic R-module. Sup p ose that M is weakly R 2 -quasi-injective. Let us p roceed by induction. Assume that M is weakly R n – 1 –quasi-injective and let x1, x2, , xn . By p rop osition 5.6, there exists a submodule Rx   such that x1, x2, , xn – 1  Rx ≈ M . But M is weakly R 2 -quasi-injective, so t here exists a submodule X of  such that X ≈ M and x, xn  X. Hence x1, x2, , xn X ≈ M . Therefore M is weakly quasi-injective (by corollary 5.5). 5.9 Corollary A cyclic R-module is weakly R n -quasi-injective if and only if it is weakly R 2 -quasi- injective. Proof: follows from theorem 5.3 and p rop osition 5.8. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.23 (1) 2010 Re ferences 1. Jain, S. K. and Lopes Permonth, S. R., (1990), "A Survry on the Theory of Weakly - Injective M odules", Comp utational Algebra, M arcel Dekker, 205-232. 2. Salih, M ., (1999), "A note on Tightness", M ath. Dep . Birzeit University , P.O.Box 14, Glasqow M ath. J., 41: 43-44. 3. Jain, S. K. and Lopez-P, S. R., (1990), "Rings whose Cy clics are Essentially Embeddable in Projective M odules", J. of Algebra, 128(1): 208-220. 4. Kasch F., (1982), "M odules and Rings", Academic p ress, London, NewYork. 5. M ijbas A. S., (1997), "Quasi-Dedekind M odules", Ph.D. T hesis, University of Baghdad. 6. Faith II, C., (1976), "Algebra, Rings Theory", Sp ringer-Verlay, Berlin Heidelberg, New York. 7. Somchit Chotchasithit, (2002), "When is Quasi-p -Injective M odule Continuous, South east Asian Bulletin of M athematics, 26: 391-394. 8. Goodearl, K. R., (1976), "Ring Theory", M arcel Dekker, New York. 9. Nicholson, W. K. and Desale, G., (1981), "Endoprimitive Rings", J. Algebra, 70: 548- 560. 2010) 1( 23المجلد مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة اغماریة نسبیة ضعیفة - مقاسات شبه ف، لیلى سلمان محمود علي سبع مجباس، كریم صبر خل جامعة بغداد ، بن الهیثمكلیة التربیة ا ،قسم الریاضیات جامعة تكریت ،كلیة علوم الحاسبات والریاضیات ،قسم الریاضیات جامعة االنبار ،كلیة العلوم ، قسم الفیزیاء الخالصة أعطینا هذا البحث اعماماً للمفاهیم . Rمقاسا احادیاً على Nو Mحلقة تبادلیة بمحاید وكل من Rلتكن ضعیف N –اغماري –مقاس شبه Mأسمینا . اغماریة نسبیة ضعیفة واحكام االغالق النسبیة واغماریة ضعیفة للمقاسات ,f  Hom(N اذا كان لكل  ان ، إذ من Xیوجد مقاس جزئي ، Mاغماري للمقاس –الغالف الشبه  ، إذ ( f (N)  X ≈ M . .Mیمكن ان یغمر في یغمر في Nمن N / Kاذا كان كل كسر N -محكم االغالق-مقاس شبه Mواسمینا ة Nلكل مقاس منته التولد ضعیف N –اغماري –شبه Mاغماري ضعیف اذا كان -مقاس شبه Mبینما اسمینا على الحلق R . ً عن ذلك عممنا بعض الخواص للمقاسات االغماریة فضال– N االغالق الضعیفة والمحكمة– N واالغماریة الضعیفة االغماریة الضعیفة على –وشبه ، N –االغالق المحكمة –وشبه ، الضعیفة N –االغماریة –الى المقاسات شبه .راسة العالقة بین هذه المفاهیموقمنا بد. التوالي