The Dual Notion of St-Polyform Modules

In the year 2018, the concept of St-Polyform modules was introduced and studied by Ahmed, where a module M is called St-Polyform, if for every submodule N of M and for any homomorphism f:N M, kerf is St-closed submodule in N. The novelty of this paper is that it dualizes this class of modules to a form that we denote as CSt-Polyform modules. Accordingly, some results that appeared in the original paper are dualized. For example, we prove that in the class of hollow modules, every CSt-Polyform module is Coquasi-Dedekind. In addition, several important properties of CSt-Polyform module are established, while further characterization of CSt-Polyform is provided. Moreover, many relationships of CStPolyform modules with other related concepts are considered, such as the copolyform, epiform, CSt-semisimple, -nonsingular modules, while some others will be introduced, such as the non-CSt-singular and G. Coquasi-Dedekind modules.


Introduction
Throughout this paper, all rings are commutative with non-zero identity elements and all modules are unitary left R-modules. The aim of this paper is to dualize the concept of St-

ISSN: 0067-2904
Polyform modules which was first studied by Ahmed [1]. For the sake of completeness, we begin with some definitions and notations that will be followed in this paper. A non-zero submodule N of M is called essential (semi-essential) if N P(0) for each non-zero submodule (prime submodule) P of M [2,3]. A submodule P of M is called prime, if whenever rmP for rR and mM, then either mP or r(P:M). A  Hadi and Ibrahiem introduced P-small submodules as an extension to the concept of small submodules, where a proper submodule N of an R-module M is called P-small (simply N M), if N+P≠M for every prime submodule P of M [6]. A generalization of coessential submodules appeared in another study [7], where a submodule L is called cosemi-essential of N in M, if . A submodule N is called coclosed in M (simply N M), if N has no proper coessential submodule in M [8]. Ahmed (3.4), (3.10) and (3.17). In addition, we determine a commutative ring having a faithful CSt-Polyform module, see Proposition (2.8). Moreover, the relationships of CSt-Polyform module with other related concepts are considered; see the results (3.2), (3.4), (3.8), (3.10), (3.15), (3.16), (3.21) and (3.23).

CSt-Polyform Modules
In this section, we dualize the class of St-Polyform and call it CSt-Polyform module. In the following, we give some examples and remarks. Before that, a submodule N of an R-module M is called corational, if (M,N/K)=0 for all submodule K of N, and an Rmodule M is called copolyform, if every small submodule of M is corational [9]. Examples and Remarks (2.2) 1. Every CSt-Polyform module is copolyform, since every CSt-closed submodule is coclosed [7]; hence, the result follows directly from the definition of CSt-Polyform module. 2. The converse of (1) is not true in general; for example, the Z-module Z is copolyform. In fact, the only small submodule of Z is (0), which is corational in Z. On the other hand, Z is not CSt-Polyform. To show that, consider the submodule (4) of Z. Let f: ZZ/(4) be a homomorphism. Note that Z/(4)  Z 4 and (Z,Z 4 )=0. On the other hand, (0) Z 4 [7], thus Z is not CSt-Polyform. 3. Every simple module is CSt-Polyform module. In fact, the only proper submodule of any module M is (0), so for all non-zero homomorphism f:M M/(0), f(M) is either zero, which is a contradiction, or M. Since M is not P-small submodule of itself, therefore M is CSt-Polyform. 4. For each prime number P, is not CSt-Polyform Z-module, since it is not copolyform, such as Z 4 , Z 9 , Z 25 , Z 49 . In fact, is a small submodule of but not corational in , since Hom Z ( )0. 5. Z 6 is a CSt-Polyform Z-module; see Example (3.5). 6. Z 4 is not CSt-Polyform Z-module, since Z 4 is not copolyform, so by (2.2)(1), Z 4 is not CSt-Polyform.

Remark (2.3):
If a submodule N of M is CSt-Polyform module, and N is essential submodule of M, then M is not necessarily CSt-Polyform; for example, suppose that M= and N= p . Note that p is CSt-Polyform Z-module, because p is simple for each prime number p, see Remark (2.2)(3). On the other hand, N is essential in , but is not CSt-Polyform Z-module. Now, we provide conditions under which the converse of Remark (2.2)(1), will be satisfied. Before that, a module M is called almost finitely generated, if M is not finitely generated and every proper submodule of M is finitely generated [6]. is copolyform module [10]. Also, it is almost finitely generated. Hence, by Proposition (2.5), is a CSt-Polyform module. Following [11], an R-module M is called multiplication, if for each submodule N of M, there exists an ideal I if R such that N=IM. Proposition (2.7): In the class of multiplication (or finitely generated or almost finitely generated modules), CSt-Polyform coincides with the class of copolyform modules. Proof: The difference between CSt-Polyform and copolyform concepts are depend on the difference between CSt-closed and coclosed submodules. Beside that the last two classes are coincide under multiplication, finitely generated, and almost finitely generated conditions as we can see in [7] and Lemma (2.4) (2). For that reason CSt-Polyform and copolyform modules are coincide under the same conditions.
Recall that a ring R is called semiprime, if for each element rR, whenever r 2 =0, then r=0 [2, P.2]. The CSt-Polyform R-module can be used as a useful condition in the following proposition. Proposition (2.8): If a commutative ring R has a faithful CSt-Polyform R-module, then R is semiprime ring. Proof: Suppose that R is a commutative ring that has a faithful CSt-Polyform module, say M. For each non-zero element xR, define f x : MM by f(m)=xm mM. We can easily show That is, (xM/x 2 M)+ (N/x 2 M)=(M/x 2 M), which implies that xM+N=M. We should prove that xMN; let xtxM and tM. Since xM+N=M, then t=xy+n, where yM and nN. By multiplying the two sides by x, we get xt=x 2 y+xn. But x 2 MN, therefore x 2 yN, also xnN, thus xtN, that is xM=x 2 M for all non-zero xR. To prove that R is semiprime, let rR with r 2 =0. Note that r 2 M=rM=0. This implies that r M, but M is faithful, thus r=0. This completes the proof. The following theorem gives another characterization of CSt-Polyform module. Before that, we need to give the following lemma. The following result can be concluding from Proposition (2.11). Also, it can be proved as follows, before that we need to give the following lemma.

CSt-Polyform modules and other related concepts
This section deals with the relationships of CSt-Polyform modules with other related concepts, such as epiform, CSt-semisimple, non-CSt-singular, -non-CSt-singular, and Coquasi-Dedekind modules.
Following [10], a non-zero module M is called epiform, if each non-zero homomorphism f: MM/N with N is a proper submodule of M, which is an epimorphism. For example, the Z-module is epiform [10].

Remark (3.1):
It is clear that every epiform module is CSt-Polyform. In fact, f(M)= M/N in the definition of epiform, which is CSt-closed in itself [7], so it is a CSt-Polyform module. The converse is not true in general; for example, Z 6 is CSt-Polyform Z-module, as we showed in Example (2.2)(5), but not epiform [10].
Under certain conditions, CSt-Polyform module can be epiform; before that, an R-module M is called prime hollow (simply P r -hollow) if each proper prime submodule of M is small [13].  [13]. This implies that f(M) is P-small submodule of M/N. But this is a contradiction, thus f(M)=M/N and, consequently, M is an epiform module. The converse is clear. Note that Theorem (3.2) represents an analogue of that appeared in [10] for copolyform modules.
Recall that a module M is called CSt-semisimple, if every submodule of M is CSt-closed [7]. Before giving the next result, we need the following. We can summarize the relations mentioned in the previous results and argument by the following implications; before that, a module M is called -noncosingular, if for every non-zero module N and every non-zero homomorphism f: MN, Im f is not small submodule in N [15]. It is clear that every noncosingular module is -noncosingular. non-CSt-singular -non-CSt-singular noncosingular -noncosingular non-CSt-singular CSt-Polyform non-CSt-singular Copolyform CSt-Polyform -non-CSt-singular noncosingular -noncosingular Following [16]; An R-module M is called fully prime module, if every proper submodule of M is prime. The following theorem gives some relations of CSt-Polyform with other modules under the class of fully prime modules. Before that, we need the following lemma. The necessity follows by [7]. For the converse, let N be a submodule of M such that N+L=M, where L M. If L is a proper submodule of M, then by assumption, L is prime. This implies that N is not P-small submodule, but this is a contradiction, thus N=L, hence N M.  .21) is not true in general; for example, the Zmodule Z is G. Coquasi-Dedekind. In fact, every non-zero endomorphism of Z is not P-small submodule of M, while Z is not CSt-Polyform, see Example (2.2)(2).
In the following theorem, we use a condition under which the converse of Proposition