n−Primary Submodules Small Monoform Modules Inaam, M.A.Hadi Hassan K. Marhun* Department of Mathematics/ College of Education for Pure Science (Ibn Al- Haitham),University of Baghdad Received in : 5 May 2014 , Accepted in : 22 June 2014 Abstract Let R be a commutative ring with unity, let M be a left R-module. In this paper we introduce the concept small monoform module as a generalization of monoform module. A module M is called small monoform if for each non zero submodule N of M and for each f ∈ Hom(N,M), f ≠ 0 implies ker f is small submodule in N. We give the fundamental properties of small monoform modules. Also we present some relationships between small monoform modules and some related modules. Key Words: Monoform module, small monoform module, small submodule, prime module, small prime module, uniform module, non singular module, quasi-Dedekind module. * This paper is a part of the thesis submitted by the second author. 229 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Introduction Throughout this article, R denotes a commutative ring with identity, and modules are unitary left R-module. We write N ≤ M to denote that N is a submodule of M. A proper submodule L of M (L < M) is called small in M (denoted by L ≪ M) if, for every proper submodule K of M, L + K ≠ M. A submodule N of M is called essential in M (denoted by N ≤e M) if N ∩ K ≠ 0 for each K ≤ M, K ≠ 0, [1]. An R-module M is called monoform module if for each non zero submodule N of M and for each f ∈ Hom(N,M), f ≠ 0 implies ker f = 0 (i.e. f is monomorphism, [2]). Equivalently M is monoform R-module if and only if M is uniform and prime module [3,theorem(2.3)], where M is uniform if every nonzero submodule N of M, N ≤e M, [1]. M is called prime R-module if annRM = annRN, for each nonzero submodule N of M, [4], where annRM = {r ∈ R: rM = 0}. In this paper, we introduce the concept small monoform as a generalization of monoform module, where M is called small monoform if for each N ≠ 0, N ≤ M, f ∈ Hom(N,M), f ≠ 0 implies ker f ≪ N. It is clear that every monoform module is small monoform, however the converse is not true (see Rem. and Ex. 1.2 (1)). We give many properties of small monoform. Also we see that under certain class of modules small monoform and monoform modules are equivalent. Moreover, we introduce many relationships between small monoform module and other related modules such as small quasi-Dedekind modules, quasi-Dedekined module, compressible modules. 1- Main Results Definition 1.1: Let M be an R-module. M is called small monoform if for each non-zero submodule N and for each f ∈ Hom(N,M), f ≠ 0 implies ker f ≪ N. Remarks and Examples 1.2: (1) It is clear that every monoform module is small monoform. However the converse is not true in general for example: The Z-module Z4 is not monoform because there exists Z-homomorphism, f : Z4 → Z4 such that f ( x ) = 2 x for each x ∈ Z4 and ker f = < 2 > ≠ ( 0 ). But Z4 is small monoform Z-module since the only non zero submodule of Z4 are < 2 > and Z4 and the only non zero Z-homomorphism from < 2 > in Z4 is the inclusion mapping i and ker (i) = < 0 >. Also there are three nonzero homomorphism from Z4 in to Z4 which are f1 = identity mapping, f2( x ) = 2 x and f3( x ) = 3 x . Hence ker( fi) ≪ Z4, ∀ i = 1,2,3. (2) It is clear that every chained module is small monoform, where an R-module is called chained module if the lattice of submodules is linearly ordered. In particular, each of the Z-module, p Z ∞ , Z4, Z8, Z16, … is small monoform. (3) The epimorphic image of small monoform module is not necessarily small monoform, for example Z as Z-module is monoform since Z is uniform and prime. But π:Z→ Z/12Z ≅ Z12, where π is the natural projection. However Z12 as Z-module is not small monoform, since 230 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 if we take N = < 2 > and f : N → Z12 defined by f ( x ) = 2 x for each x ∈ N, ker f = {0 , 6} ⊆ N. But {0 , 6} + {0 , 4, 8} = N. Thus ker f  N. (4) Every non zero submodule of small monoform module is small monoform module. Proof: Let M be a small monoform R-module and let N ≤ M. For any K ≤ N, K ≠ 0, let f : K → N, f ≠ 0. K N M , 0f i i f→ → ≠ . But ker(i ∘f ) = ker f , hence ker f ≪ K. Thus N is small monoform. Recall that: If M is an R-module, then M is an R -submodule of M, where R = R/ann M by using the definition (r + R ann M)x = rx, for each x ∈ M. Hence every R-submodule of M is an R -submodule of M, and conversely. (5) Let M be an R-module. Then M is small monoform R-module if and only if M is small monoform R -module Proof: (⇒) Let N be an R -submodule of M, let f : N → M, f ≠ 0 be R -homomorphism. It is clear that N is R-submodule of M. To show that f is an R-homomorphism. Let r ∈ R, f (rx) = f [(r + annM)x] = (r + annM) + f (x) since f is an R -homomorphism = r f (x) Thus f is an R-homomorphism. But M is small monoform, so ker f is small R-submodule of N. Hence ker f is small R -submodule of N. The proof of the converse is similarly. Remark 1.3: Let M be a semisimple R-module. Then the following statements are equivalent: (1) M is small monoform. (2) M is monoform. (3) M is simple. Proof: (1) ⇒ (2) Let N ≤ M, let f : N → M, f ≠ 0. Since M is small monoform, then ker f ≪ N. But M is semisimple, so N is semisimple and hence N has only small submodule namely (0). Thus ker f = (0) and so M is monoform. (2) ⇒ (1) It is clear by (Rem. and Ex. 1.2(1)). (2) ⇒ (3) Let x ∈ M, x ≠ 0. Since M is semisimple, then is a direct summand of M. So ⊕ K = M, for some K ≤ M. But M is monoform, so for each homomorphism f : → M, f ≠ 0, ker f = 0. Define g : M → M, by g(rx + K) = f (rx). We can show that g is well-defined as follows: Let r1x + k1 = r2x + k2 where r1, r2 ∈ R, k1, k2 ∈ K (r1 – r2)x = k2 – k1 ∈ ∩ K = (0). Hence (r1 – r2)x = 0 = k2 – k1. Thus implies r1x = r2x and k1 = k2. Thus f (r1x) = f (r2x) and g(r1x + k1) = g(r2x + k2). Now let rx + k ∈ ker g, then g(rx + k) = f (rx) = 0. It follows that ker g = ker f ⊕ K = 0 ⊕ K = K. But ker g = 0, so K = 0. 231 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Thus = M and therefore M is simple. (3) ⇒ (2) If M is simple, then M has only two submodules (0), M. So that for each f : M → M, f ≠ 0 ker f ≤ M, hence ker f = 0. Thus M is monoform. Recall that an R-module M is called free if it has a basis, [1]. Theorem 1.4: Let M be a free Z-module. Then M is small monoform if and only if M is monoform. Proof: (⇒) Let N ≤ M, N ≠ 0, let f : N → M, f ≠ 0. Since M is small monoform implies ker f ≪ N. But M is a free Z-module implies N is a free Z-module, [5,Corollary (5.5.3)]. So, N has only (0) small submodule. Thus ker f = 0; that is M is monoform. (⇐) It is clear by 1.2(1). The following proposition gives a characterization of small monoform module under the class of Noetherian modules. Proposition 1.5: Let M be a non zero Noetherian R-module. Then M is small monoform if and only if every non zero 3-generated submodule of M is small monoform. Proof: (⇒) It is clear. (⇐) suppose every non zero 3-generated submodule of M is small monoform. Let N ≤ M, N ≠ 0 and let f ∈ Hom(N,M), f ≠ 0. To prove ker f ≪ N. If ker f = (0) then ker f ≪ N. If ker f ≠ (0), let x ≠ 0 and x ∈ ker f. Let y ∈ N and let f (y) = z. Put L = , L is 3-generated submodule of M. By hypothesis, L is small monoform, let H = . Let g = f H: H → L. Hence ker g ≪ H ≤ N, since L is small monoform. This implies ker g ≪ N. But x ∈ ker g, so that ⊆ ker g ≪ N. Thus ≪ N. Since M is Noetherian, ker f is finitely generated. Hence ker f = Rx1 + Rx2 + … + Rxi = for x1, …, xn ∈ M. Since ≪ N for each i = 1, …, n. So n i i 1 ker Rxf = = ∑ ≪ N. Thus M is small monoform. Recall that an R-module M is called uniform if every non zero submodule is eesential, [1]. Recall that an R-module M is called quasi-Dedekind if for each N ≤ M, N≠ 0, Hom( M N ,M)=0, that is every nonzero submodule N of M is quasi-invertible, [6]. Recall that an R-module M is called small quasi-Dedekind if for each f ∈ End(M), f ≠ 0, ker f ≪ M. Equivalently M is small quasi-Dedekind if Hom(( M N ,M) = 0 for each N  M [7], where End(M) = set of all homomorphism from M to M. 232 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Let A be a submodule of an R-module M. A relative complement for A in M is any submodule B of M which is maximal with respect to the property A ∩ B = 0 [8,p.17]. Proposition 1.6: Let M be an R-module if M is small monoform, then M is uniform and M is small quasi- Dedekind. Proof: By [7, Rem. and Ex. (3.2.9),p.109] M is small quasi-Dedekind, let N ≤ M, N ≠ (0). If N ≤e M nothing to prove. Suppose N ≰e M, then there exists (H ≤ M) such that H is a relative complement of N. Hence N ⊕ H ≤e M by [8,proposition 1.3,p.17]. Define f : N ⊕ H → M by f (a + b) = a for each a + b ∈ N ⊕ H. Then ker f = (0) ⊕ H, but M is small monoform, so ker f = (0) ⊕ H ≪ N ⊕ H and this implies H ≪ H (which is impossible) unless H = (0) and hence N ≤e M. Thus M is uniform. Corollary 1.7: Let M be an R-module. If M is small monoform, then M is uniform and R ann M = R ann N for each N  M. Proof: By proposition 1.6, M is uniform. Also M is small quasi-Dedekind, hence for each N  M, N is a quasi-invertible [7,Th. 3.1.3,p.95]. Thus R ann M = R ann N for each N  M [6, proposition 1.4,p.7] Recall that an R-module Z(M) = {x ∈ M, annR(x) ≤e R} is called a singular submodule of M. If Z(M) = M, then M is a singular module. If Z(M) = 0, then M is called a non singular module, [8,p.31]. Proposition 1.8: Let M be a non singular R-module. Then M is small monoform implies M is quasi- Dedekind. Proof: Let N ≤ M. Since M is small monoform implies M is uniform (by proposition 1.6). Hence N ≤e M, but N ≤e M and M is a non singular implies M N is singular [8,proposition 1.21,p.32]. Hence Hom( M N ,M) = 0 [8,Exc. 1,p.33]; that is N is quasi- invertible. Thus M is quasi-Dedekind. Note 1.9: The condition M is nonsingular in Proposition 1.8 is necessary. For example Z4 as Z-module is small monoform, but is not quasi-Dedekind. Also Z4 is not a nonsingular Z- module, since Z(Z4) ≠ (0) (in fact Z(Z4) = Z4). It is known that: A ring R is semisimple implies every R-module is a non singular. Hence we get the following corollary. 233 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Corollary 1.10: Let R be a semisimple ring, let M be an R-module, then M is small monoform implies M is quasi-Dedekind. Proof: Since R is semisimple, M is nonsingular. Hence the result follows by proposition 1.8. Recall that an R-module M is called a prime R-module if R ann M = R ann N for every non zero R-submodule N of M, [4]. Corollary 1.11: Let M be a non singular small monoform, then M is prime. Proof: By Proposition 1.8, M is small monoform and non singular implies M is quasi-Dedekind. Thus M is prime [6,proposition 1.7, p.26]. Recall that an R-module M is called fully retractable, if for every non zero submodule N of M and every non zero element g ∈ HomR(N,M) we have HomR(M,N)g ≠ 0, [9]. Proposition 1.12: Let M be an R-module such that M is fully retractable and for each N ≤ M, N ≠ (0), N is small quasi-Dedekind, then M is small monoform. Proof: Let N ≤ M, f :N → M, f ≠ 0. Since M is fully retractable, then there exists g : M → N, g ≠ 0. Consider N M Nf g→ → . By M fully retractable, g ∘ f ≠ 0. Since N is small quasi-Dedekind, ker (g ∘ f ) ≪ N. But ker f ⊆ ker (g ∘ f ) and this implies ker f ≪ N. Thus M is small monoform. Recall that an R-module M is called a qusi-injective R-module if for each monomorphism h: N → M, where N is any R-submodule of M and any homomorphism ϕ: N → M, there is a homomorphism ψ: M → M such that ψ∘h= ϕ i.e. the following diagram is commutative, [10,p.22]. hN M→ M Recall that A submodule N of M is called coclosed if whenever K ≤ N and N M K K  implies K = N, [11]. We prove the following: ψ ϕ 234 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Proposition 1.13: Let M be a quasi-injective R-module and every submodule of M is coclosed then M is small quasi-Dedekind if and only if M is small monoform. Proof: (⇒) Let N ≤ M, N ≠ (0), let f ∈ Hom(N,M), f ≠ 0 consider the following diagram N Mi→ M Since M is quasi-injective, then there exists g ∈ End(M) such that g ∘ i = f . Hence g(n) = f (n) for each n ∈ N, which implies ker f ≤ ker g. But M is small quasi-Dedekind, so ker g ≪ M. Thus implies ker f ≪ M. But every submodule of M is coclosed, then N is coclosed. Thus ker f ⊆ N and ker f ≪ M which implies ker f ≪ N, [12, Lemma 1.1]. Therefore M is small monoform. (⇐) It is clear. Under the class of non singular modules, we have the following: Proposition 1.14: Let M be a non singular R-module. Then the following statements are equivalent: (1) M is small monoform. (2) M is uniform quasi-Dedekind (3) M is uniform prime. (4) M is uniform. (5) M is monoform. Proof: (1) → (2) By Proposition 1.6 M is uniform. But M is small monoform and non singular implies M is quasi-Dedekind by Proposition 1.8. (2) → (3) It is follows by [6, Proposition 1.7, p.26]. (3) → (4) It is clear. (4) → (5) It follows by [3, Theorem 2.2]. (5) → (1) It is clear by 1.2(1). Recall that an R-module M is called compressible if for each N ≤ M, N ≠ 0 M can be embedded in M (i.e. there exists f : M → N, f is monomorphism), [13]. Consider the following statement (∗): (∗) Let M be an R-module such that R M ann N ⊈ annM, for each N ≤ M, N ≠ 0. We prove the following: Proposition 1.15: ∃ g f 235 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Let M be a nonsingular R-module such that M satisfies (∗). Then the following statements are equivalent (1) M is small monoform. (2) M is quasi-Dedekind. (3) M is prime. (4) M is compressible. (5) M is monoform (6) M is uniform (7) EndR(M) is an integral domain. (8) R/annRM is an integral domain. (9) annRM is a prime ideal in R. Proof: (1) → (2) By Proposition 1.8. (2) → (3) It is follows by [6, proposition 1.7, p.26]. (3) ↔ (4) ↔ (5) It is follows by [14, proposition 1.7]. (5) ↔ (6) It is follows by [3, theorem 2.2]. (5) → (1) It is clear. i.e. (1) ↔ (2) → (3) ↔ (4) ↔ (5) ↔ (6) (3) ↔ (9) It is follows by [14, proposition 1.9]. (4) ↔ (7) ↔ (8) It is follows by [14, theorem 2.5]. i.e. (6) ↔ (3) ↔ (9) ↔ (4) ↔ (7) ↔ (8). Thus all statement (1) through (9) are equivalent. Corollary 1.16: Let M be a multiplication non singular R-module. Then the statements from 1 to 9 in proposition 1.15 are equivalent. Proof: It follows directly by proposition 1.15, since every multiplication module satisfies (∗). Recall that an R-module M is called retractable if HomR(M,N) ≠ 0 for all 0 ≠ N ⊆ M, [15]. Proposition 1.17: Let M be retractable and nonsingular R-module, then the following statements are equivalent: (1) M is monoform. (2) M is uniform. (3) M is small monoform. (4) M is compressible. Proof: (1) ↔ (2) It follows by [3, theorem 2.2]. (1) → (3) It is clear by 1.2(1). (3) → (2) It follows by Proposition 1.6. (2) → (4) It follows by [9, Proposition 1.7]. (4) → (1) It follows by [3,corollary 2.5]. Recall that an R-module M is called small prime if R ann M = R ann N for each N ≪ M, [16]. 236 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Proposition 1.18: Let M be small monoform and small prime R-module. Then M is monoform. Proof: Since M is small monoform then M is uniform by Proposition 1.6. Also R ann M = R ann N for each N  M by proposition 1.7. But by hypothesis M is small prime, so for each N ≪ M, N ≠ (0), R ann M = R ann N. Thus ann M = ann N for each N ≤ M, N ≠ (0), that is M is a prime R-module. But M is uniform and prime implies M is monoform [3,theorem 2.3]. Under the class of finitely generated modules, we have the following result. Corollary 1.19: Let M be a finitely generated R-module, then the following statements are equivalent: (1) M is monoform. (2) M is uniform prime. (3) M is quasi-Dedekind. (4) M is small monoform and small prime. (5) M is is compressible. Proof: (1) ↔ (2) It follows by [3, Theorem 2.3]. (2) ↔ (3) It follows by [6, Corollary 3.13]. (1) → (4) It is clear. (4) → (1) It follows by Proposition 1.18. (5) ↔ (1) It follows by [3,Lemma 1.9 and Theorem 2.3]. Next we turn our attention to direct sum of small monoform R-modules Remark 1.20: M = M1 ⊕ M2, M1 and M2 submodule of M, M is small monoform. Then M1 and M2 are small monoform. But the converse is not true in general. Proof: (⇒) It is clear by Rem. and Ex. 1.2 (5). Now, consider the following example: Let M = Z4 ⊕ Z4 as Z-module, Z4 as Z-module is small monoform (by Remarks and Examples 1.2 (1)), let N = Z4 ⊕ 2< > . Let f : Z4 ⊕ 2< > → Z4 ⊕ Z4 defined by (x, y) (x, 2y)f = , 4ker {(0, 0), (0, 2)} (0) 2 Z (2)f = = ⊕ < > ⊕ since 4 40 2 (Z 0 ) Z 2< > ⊕ < > + ⊕ < > = ⊕ < > Thus the direct sum of small monoform modules need not be small monoform. Recall that an R-module M is called fully stable if for each N ≤ M. N is stable; that is for each f : N → M, f is R-homomorphism, f (N) ⊆ N, [17]. Now we show that under certain condition, the direct sum of small monoform is small monoform. 237 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Theorem 1.21: Let M be a fully stable R-module, such that M = M1 ⊕ M2, M1, M2 ≤ M and for each R- homomorphism f : N1 ⊕ N2 → M, f ≠ 0 implies f (N1) ≠ 0, f (N2) ≠ 0 (i.e. f /N1 ≠ 0, f /N2≠0). Then M1 and M2 are small monoform if and only if M is small monoform. Proof: (⇒) Let N ≤ M, N ≠ (0), f :N → M, f ≠ 0, to prove ker f ≪ N. Snce M is fully stable, every submodule of M is stable so, N is stable and this implies N = (N ∩ M1) ⊕ (N ∩ M2) by [17,Prop.4.5,p.29]. Consider 1 11 1(N M ) N M M i f ρ∩ → → → , 2 22 2(N M ) N M M i f ρ∩ → → → Where i1, i2 are inclusion mappings and ρ1, ρ2 are projection mappings. Then ρ1∘f∘i1: (N ∩ M1) → M1, ρ2∘f∘i2: (N ∩ M2) → M2, let N1 = N ∩ M1 N2 = N ∩ M2. By hypothesis f /N1 ≠ 0, so there exists n1 ∈ N ∩ M1, n1 ≠ 0, f (n1) ≠ 0 and f /N2 ≠ 0, so there exists n2 ∈ N ∩ M2, n2 ≠ 0, f (n2) ≠ 0. On the otherhand f ∘ i1: (N ∩ M1) → M implies f ∘ i1(n1) = f (n1) ≠ 0, f ∘ i2:(N ∩ M2) → M implies f ∘ i2(n2) = f (n2) ≠ 0. Thus implies f ∘ i1(N ∩ M1) ⊆ N ∩ M1, since N1, N2 are stable. Hence f (N∩ M1) ⊆ N ∩ M1. Similarly f (N ∩ M2) ⊆ N ∩ M2. But f (n1) ∈ N ∩ M1 and f (n1) ≠ 0 and, so that (ρ1∘f∘i1)(n1) = f (n1) ≠ 0. Similarly (ρ2∘f∘i2)(n2) = f (n2) ≠ 0. Thus ρ1∘f∘i1 ≠ 0, ρ2∘f∘i2 ≠ 0. Since M1, M2 are small monoform, then ker(ρ1∘f∘i1)⊕ker (ρ2∘f∘i2)≪(N ∩ M1)⊕(N ∩ M2)=N. Let 1 2x n n′ ′= + ∈ker f, where 1n′∈ N ∩ M1, 2n′ ∈ N ∩ M2, 1 2(n ) (n ) 0f f′ ′+ = . Hence 1 2(n ) (n )f f′ ′= − ∈ (N ∩ M1) ∩ (N ∩ M2) = (0), it follows 1 2(n ) 0 , (n ) 0f f′ ′= = . This implies ρ1∘f∘i1( 1n′ )=ρ1∘f ( 1n′ ) = ρ1( f ( 1n′ ))= f ( 1n′ )= 0 ρ2∘f∘i2( 2n′ )=ρ2∘f ( 2n′ ) = ρ2( f ( 2n′ ))= f ( 2n′ )= 0. Hence 1 2x n n′ ′= + ∈ ker(ρ1∘f∘i1)⊕ker (ρ2∘f∘i2) ≪ N. So that ker f ⊆ ker(ρ1∘f∘i1) ⊕ ker (ρ2∘f∘i2) ≪ N. Thus ker f ≪ N. Therefore M is small monoform. (⇐) It is clear by remarks and examples 1.2 (4). References 1. Kasch, F., (1982), Modules and Rings, Academic Press, Inc-London. 2. Zelmanowitze, J.M., (1986), Representation of Rings With Faithful Polyform Modules, Comm. in Algebra, Vol.14, No.6, pp.1141-1169. 3. Smith, P.F., (2006), Compressible and Related Modules, in Abelian Groups, Rings, Modules and Homological Algebra, eds, P.Goeters and O.M.G. Jend (Chapman and Hull, Boca Raton), pp.1-29. 4. Desal, G. and Nicholson, W.K., (1981), Endoprimitive Rings, J.Algebra, Vol. 70, pp.548-560. 5. Hazewinkel, M., Gubareni, N. and Kirichenka, V.V., (2004), Algebras, Rings and Modules, Vol.1, Kluwer Acadmic Puplishers. 6. Mijbass, A.S., (1997), Quasi-Dedekind Modules, Ph.D. Thesis, University of Baghdad, Iraq. 238 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 7. Ghaw Al-Jubory, T.Y., (2010), Some Generalizations of Quasi-Dedekind Modules, M.Sc. Thesis, University of Baghdad, Iraq. 8. Goodearl, K.R., (1976), Ring Theory, Non Singular Ring and Modules, Marcel Dekker, Inc. New York and Basel. 9. Rodrigues, V.S. and San't Ana, A.A., (2009), A Problem Due Zelmanowitz, Algebra and Discrete Math., pp.1-9. 10. Faith, C., (1967), Lectures on Injective Modules and Qutient Rings, Springer-Verlage, Berlin, Heidelberg, New York. 11. Golan, J.S., (1971), Quasi-Semiperfect Modules, Quart, J.Math., Oxford (2), Vol.22, pp.173-182. 12. Keskin, D., (2000), On Lifting Modules, Comm. In Algebra, Vol.28, pp.3427-3440. 13. Zelmanowitze, J.M, (1976), An Extension of the Jacobsin Density Theorem, Bull. Amer.Math.Soc., Vol.82, No.4, pp.551-553. 14. Ahmed, A.A., (1995), A note on Compressible Module, Abhath Al-Yarmouk, Irbid, Jordan, Vol.4, No.2, pp.139-148. 15. Roman, C.S., (2004), Baer and Quasi Baer Modules, Ph.D. Thesis, M.S.Graduate, School of Ohio, State University. 16. Mahmood, L.S., (2012), Small Prime Modules and Small Prime Submodule, Journal of Al-Nahrain University Science, Vol.15, No.4), pp.191-199. 17. Abbas, M.S., (1990), On Fully Stable Modules, Ph.D. Thesis, University of Baghdad, Iraq. 239 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 ذات الصیغة المتباینة الصغیرةالمقاسات إنعام محمد علي ھادي *حسن خلیف مرھون ، جامعة بغداد (ابن الھیثم)كلیة التربیة للعلوم الصرفة، قسم الریاضیات 2014حزیران 22،قبل البحث في 2014ایار 5استلم البحث خالصةال . قدمنا في ھذا البحث مفھوم المقاسات ذات الصیغة Rمقاسا ً أیسر على Mحلقة إبدالیة ذات محاید ولیكن Rلتكن مقاس ذي صیغة متباینة صغیرة اذا كان لكل مقاس Mللمقاسات ذات الصیغة المتباینة. یسمى ا ًالمتباینة الصغیرة تعمیم . Nمقاس جزئي صغیر في ker fیؤدي الى f ≠ 0و f ∈ Hom(N,M)ولكل تشاكل Mفي Nجزئي غیر صفري أعطینا الخواص االساسیة للمقاسات ذات الصیغة المتباینة الصغیرة. كذلك قدمنا بعض العالقات بین المقاسات ذات الصیغة المتباینة الصغیرة مع بعض المقاسات المرتبطة معھا. صغیرة، مقاس جزئي صغیر ،المتباینة المقاس ذي صیغة المتباینة ، المقاس ذي صیغة ال الكلمات المفتاحیة : ولي ، مقاس أولي صغیر ، مقاس منتظم ، مقاس غیر منفرد، مقاس شبھ دیدكایند.أمقاس 240 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 1- Main Results لتكن R حلقة إبدالية ذات محايد وليكن M مقاسا ً أيسر على R. قدمنا في هذا البحث مفهوم المقاسات ذات الصيغة المتباينة الصغيرة تعميما ً للمقاسات ذات الصيغة المتباينة. يسمى M مقاس ذي صيغة متباينة صغيرة اذا كان لكل مقاس جزئي غير صفري N في M ولكل تشاكل... الكلمات المفتاحية : المقاس ذي صيغة المتباينة ، المقاس ذي صيغة المتباينة الصغيرة، مقاس جزئي صغير ، مقاس أولي ، مقاس أولي صغير ، مقاس منتظم ، مقاس غير منفرد، مقاس شبه ديدكايند.