IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 Weakly Relative Quasi-Injective Modules L. S Mahmood, A. S. Mijbass, K. S. Kalaf Department of Mathematics, Ibn Al-Haitham College of Education, University of Baghdad. Department of Mathematics, College of Computer Science and Mathematics, Tikrit University. Department of Physics, College of Science, University 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. Moreover, we generalize some properties of weakly N-injective, N-tight and weakly injective modules to weakly N-quasi-injective, N-quasi-tight and weakly quasi-injective modules respectively. The relations among these concepts are also studied. Introduction The concept of weak relative injectivity of modules was introduced originally in [1]. Since then, the study of this concept has been illustrated extensively. We introduced in this research the concept of weak relative quasi-injectivity of modules as a generalization of the concept of weak relative injectivity which motivates our principle subject of this research. This paper 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,  ) implies that f (N) is contained in This paper represents a part of Ph.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 Baghdad. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 some submodule of  which is isomorphic to M, see definition 1.1. We established some properties of such modules. We showed that the class of such modules is not closed under direct summand, see Ex. 1.6. While we could not prove or disprove that this class of modules is closed under direct sum. But we proved a special case of this, see proposition 1.7. Next we proved that this class of modules is closed under essential extension, see proposition 1.15. The second section is devoted to give some characterizations of weakly relative quasi- injective modules which are very useful in the next 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 the concept of relative quasi-tightness 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 proposition 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 section four certain relations between quasi-tight modules and compressible modules in order to relate weak relative quasi-injectivity and compressibility 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 paper, we considered those modules which are weakly quasi- injective relative to each finitely generated module we would refer to any such module as being 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 positive integer n, see 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 corollary 5.5. 2. 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  R b for all i = 1, 2, , n, see proposition 5.6. 3. A cyclic R-module is weakly quasi-injective if and only if it is weakly R 2 -quasi- injective, see proposition 5.8. Section One: Weakly Relative Quasi-Injective Modules We shall introduce in this section the concept of weakly relative qusi-injectivity of modules. The relation between weakly relative quasi-injective modules and certain types of modules are studied. Some properties of weakly relative quasi-injective modules are established. 1.1 Definition 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. SCI. 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 disprove the validity of the converse of the above remarks consider the following examples respectively: 1.3 Example i. Let M = Z2, 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) = 32 a + Z for all a  Z, then f  Hom(Z, 2  ) and f (Z) ≈ Z8 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 Proposition Let M and N be two R-modules and let I be an ideal of R such that I  annR( )  annR(N). Then 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), implies that M and N are R / I-modules. Moreover, f :N   is an R-homomorphism if and only if f is an R / I –homomorphism, and X is an R- submodule of  if and only if X is an R / I – submodule of  . Hence the details of the proof 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 example. 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. SCI. VOL.23 (1) 2010 If f  0, then f is a monomorphism, 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 implies 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 monomorphism. 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, then f (Q)  0  Q  Z  Q  Q  Q. Take X = M = Z  Q, then 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 prove that Z  Y ≈ Z  Q. Therefore 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 prove and we can not disprove that the class of weakly relative quasi- injective modules is closed under direct sum. However, we give a special case of this. 1.7 Proposition 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 respectively, such that (N)  X ≈ L and (N)  Y ≈ M . On the 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 completes 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 examples: 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. SCI. 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 Proposition 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 homomorphisms and the homomorphism  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), thus 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 explain the behavior of weakly-quasi-injectivity under homomorphism. 1.12 Proposition Let H, N and M be R-modules and let g : N  H be an epimorphism. 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 that 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 epimorphism. If M is weakly N-quasi-injective, then M is weakly g(N)-quasi-injective. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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 Proposition 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 homomorphism. 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 implies that f i = 0 and hence f (M / N) = 0. But m  M and m  N and M is essential in M , so there 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 implies that r = 0 which is a contradiction. Therefore N is quasi-invertible in M . 1.15 Proposition 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 isomorphism and i1, i2, i3 are inclusion homomorphisims. L being quasi-injective, so there exists a homomorphism  : 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 there 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. SCI. VOL.23 (1) 2010 Section Two: Characterizations of Weakly Relative Quasi Injective Modules 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 the concept of weakly quasi-injectivity can be given in terms of commutative diagram. 2.1 Theorem 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 composition g h where h : N  M is a homomorphism and g : M   is a monomorphism. 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 homomorphism. 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 homomorphism. Hence g : M   is a monomorphism. Now, g  h = (i  – 1)  ( f.) = i f = f which proves the "only if” part. To prove the "if” part: Let f  Hom(N, ). By hypothesis, there exists a homomorphism h : N  M and a monomorphism 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 Theorem Let M and N be two R-modules. Then M is weakly N-quasi-injective if and only if for any N-cyclic 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-cyclic submodule of . So, X ≈ N / K for some submodule K of N. Then we have: / i    where  is the natural homomorphism,  is an isomorphism and i is the inclusion homomorphism. Let f = i   . Then f  Hom(N, ), implies that there exists a homomorphism h : N  M and a monomorphism g : N   such that f = g h (by Theorem 2.1). IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 Now, g h(N) = f (N) = i (N) = i(N / K) = i(X) = X. Therefore g h(N) = X. We take L = g(N) to obtain that L is a submodule of  and L ≈ M . Moreover X = g(N)  g(M) = L. Conversely, To prove 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 Theorem Let M and N be two R-modules. Then the following statements 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 homomorphism f : N / K   , there exists a monomorphism g: M   and a homomorphism 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 homomorphism. 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 homomorphism. It can be easily shown that f is a monomorphism. Hence by (3), there exists a monomorphism g : M   and a homomorphism 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), implies that f (N)  X ≈ M, which proves (1). The following lemma is needed in order to give some applications 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 omitted. 2.5 Corollary Let K, M and N be R-modules. If M is weakly K-quasi-injective and N is K-cyclic. Then M is weakly N-quasi-injective. Proof: M being K-quasi-injective, implies that M is weakly K / L-quasi-injective for every submodule L of K (by theorem 2.3). But N is K-cyclic, 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. SCI. 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 implies 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 Theorem 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-homomorphism. 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. Therefore f (R)  X and hence M is weakly R-quasi-injective. As a special case, we shall characterize the weakly quasi-injectivity of the R-module R relative to itself. 2.9 Theorem 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- homomorphism. 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 isomorphism. 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) implies 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 implies that X  R . But R b ≈ R / annR(b) ≈ R. Moreover f (R) = {f (r) : r  R} = {r f (1) : r  R} = R a  R b. Therefore f (R)  X ≈ R. This completes the proof. We shall establish in the following theorem a general case of theorem 2.9. 2.10 Theorem 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   . Suppose 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. SCI. 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 homomorphism. Moreover if r x = 0, implies 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 monomorphism. Clearly h is an epimorphism. Hence X ≈ M and therefore M is weakly N- quasi-injective. When we weaken the conditions in theorem 2.10, we get the following result: 2.11 Proposition Let M and N be two cyclic 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   . Suppose 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 there 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 proposition 2.11, may not be true in general, consider the following example: 2.13 Example Let M = Z4, N = Z and R = Z. Then M is weakly N-quasi-injective. But M is not torsion- free R-module. On the other hand, example 1.6 shows that the condition N is cyclic in proposition 2.11, can not be dropped. Section Three: Weakly Relative Quasi-Injective Modules and Quasi- Tight Modules We introduce in this section the concept of relative quasi-tightness of modules and we study the relation of this concept with the concept of relative weakly quasi-injectivity of modules. 3.1 Definition 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. SCI. 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 Definition 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-tight R-module is N-quasi-tight and the converse is not true in general. Consider the following example: Let M = Z2, N = Z, and R = Z. Let K be a submodule of N. If N / K embeds in 2 = Z2, 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-tight. 3.4 Proposition 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 monomorphism f : N / K   . Let  : N  N /K be the natural homomorphism. Hence f   Hom(N, ), so there 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 homomorphism. Let g : X  M be an isomorphism. Then g f : N / K  M is a monomorphism. Which completes the proof. 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 property that A  B = 0, then A is called a complement of B in C, [8]. 3.6 Theorem 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 monomorphism f : N / L   , we have: i. There exists a monomorphism f : N / L  M, and ii. For every complement 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 monomorphism, M being weakly N-quasi-injective implies that M is weakly N / L-quasi-injective (by theorem 2.3) and hence there exists a homomorphism f : N / L  M and there exists a monomorphism  : 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 complement 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. SCI. VOL.23 (1) 2010 is a monomorphism, so y = f (z), implies 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 monomorphism. By (i), there exists a monomorphism 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. Then  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 monomorphism. Therefore  is extended to a monomorphism : 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 monomorphism. But M is N- quasi-tight, therefore there exists a monomorphism 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 complement of f (N / L) in M and M  f (N / L) = 0. If L  N, then f (N / L) is a non-zero submodule of M and since M is uniform implies that 0 is the only complement 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 proposition 3.4 and corollary 3.7. Section Four: Quasi-Tight Modules and Compressible Modules In this section, we establish some relations between relative quasi-tight modules and compressible modules in the class of quasi-injective modules. 4.1 Definition An R-module M is called compressible if for all non-zero submodules N of M, M embeds in N, [9]. In general, an R-module M is compressible if for every essential submodule N of M, M embeds in N, [4]. First, we establish the relationship between relative quasi-tight modules and compressible modules. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 4.2 Theorem 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 compressible. Proof: Assume that every submodule of M is N-quasi-tight. Let N / K be embeds in M. Hence there exists a monomorphism f: N / K  M. We have to show that N / K is compressible. 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, Suppose that every quotient of N which embeds in M is compressible. 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 monomorphism. But    = M, implies that i h : N / K  M is a monomorphism where i :    is the inclusion homomorphism. 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 compressible therefore N / K embeds in h– 1(B). On the other hand h– 1(B) ≈ B A. Thus 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 compressible. 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 that for each submodule B of A such that A / B embeds in M is compressible. But A is finitely generated implies that A / B is also finitely generated. Hence every finitely generated submodule of M is compressible. Conversely, Assume that every finitely generated submodule of M is compressible. To prove 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 hypothesis, N / K is compressible. 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 compressible. 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 compressible. Now, let A = (a) be a cyclic submodule of M for some a  M. Then A ≈ R / annR(a). So, R / annR(a) is compressible. Hence A is compressible. Conversely, Assume that every cyclic submodule of M is compressible. Because of the fact that every cyclic submodule of M can be written 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 compressible, therefore every cyclic submodule of M is R-quasi-tight (by theorem 4.2). To prove every submodule of M is weakly R-quasi- IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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. Thus 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 compressible 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 hypothesis, we get that every submodule of A is N-quasi-tight, and since A is a quasi-injective R-module, implies that N / K is compressible for every submodule K of N (by theorem 4.2). Section Five: Weakly Quasi-Injective Modules 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 Definition 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 example 1.3. 5.3 Theorem 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 positive integer n. Proof: the "only if” part is obvious. To prove the "if” part. Let N be a finitely generated R-module. We have to show that M is weakly N-quasi-injective. Suppose 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 epimorphism. Therefore, R n / ker f N. But M is weakly Rn-quasi-injective, implies that M is weakly Rn / ker f-quasi-injective (by theorem 2.3). Therefore M is weakly N-quasi –injective. 5.4 Proposition An R-module M is weakly Rn-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 homomorphism. 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 part. Conversely, IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 We have to show that M is weakly Rn-quasi-injective. Let f  Hom(Rn, ). Suppose 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 hypothesis there exists a submodule X of  such that xi  X ≈ M for all i = 1, 2, , n which implies 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 Proposition 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: Suppose that R is weakly R n-quasi-injective. Let x1, x2, , xn  R . By proposition 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 isomorphism. 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. Moreover, if r b = 0 for some r  R, implies that r = 0 and hence annR(b) = 0. Conversely, We have to show that R is weakly R n-quasi-injective. Let f  Hom(Rn, 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 Rn-quasi- injective (by proposition 5.4). The following corollary is also a consequence of theorem 5.3 and proposition 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 Proposition A cyclic R-module is weakly quasi-injective if and only if it is weakly R 2 -quasi- injective. Proof: the "only if " part is obvious. To prove the "if " part, let M be a cyclic R-module. Suppose that M is weakly R 2 -quasi-injective. Let us proceed by induction. Assume that M is weakly R n – 1 –quasi-injective and let x1, x2, , xn . By proposition 5.6, there exists a submodule Rx   such that x1, x2, , xn – 1  Rx ≈ M . But M is weakly R2-quasi-injective, so there 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 proposition 5.8. IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.23 (1) 2010 References 1. Jain, S. K. and Lopes Permonth, S. R., (1990), "A Survry on the Theory of Weakly- Injective Modules", Computational Algebra, Marcel Dekker, 205-232. 2. Salih, M., (1999), "A note on Tightness", Math. 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 Cyclics are Essentially Embeddable in Projective Modules", J. of Algebra, 128(1): 208-220. 4. Kasch F., (1982), "M odules and Rings", Academic press, London, NewYork. 5. Mijbas A. S., (1997), "Quasi-Dedekind Modules", Ph.D. Thesis, University of Baghdad. 6. Faith II, C., (1976), "Algebra, Rings Theory", Springer-Verlay, Berlin Heidelberg, New York. 7. Somchit Chotchasithit, (2002), "When is Quasi-p-Injective Module Continuous, South east Asian Bulletin of Mathematics, 26: 391-394. 8. Goodearl, K. R., (1976), "Ring Theory", Marcel 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 علي سبع مجباس، كریم صبر خلف، لیلى سلمان محمود د ، بن الهیثمكلیة التربیة ا ،قسم الریاضیات جامعة بغدا جامعة تكریت ،كلیة علوم الحاسبات والریاضیات ،قسم الریاضیات جامعة االنبار ،كلیة العلوم ، قسم الفیزیاء الخالصة م . Rمقاسا احادیاً على Nو Mحلقة تبادلیة بمحاید وكل من Rلتكن أعطینا هذا البحث اعماماً للمفاهی 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