St-Polyform Modules and Related Concepts

In this paper, we introduce a new concept named St-polyform modules, and show that the class of Stpolyform modules is contained properly in the well-known classes; polyform, strongly essentially quasiDedekind and κ-nonsingular modules. Various properties of such modules are obtained. Another characterization of St-polyform module is given. An existence of St-polyform submodules in certain class of modules is considered. The relationships of St-polyform with some related concepts are investigated. Furthermore, we introduce other new classes which are; St-semisimple and κ-non St-singular modules, and we verify that the class of St-polyform modules lies between them.


Introduction:
Throughout this paper, all rings are assumed to be commutative with a non-zero unity element, and all modules are unitary left R-modules. The notations V ≤ e U and V ≤ m U mean that V is an essential and semi-essential submodule of U respectively. A submodule V of U is called essential if every non-zero submodule of U has a non-zero intersection with V (1, P.15). A submodule V of U is called semi-essential if every non-zero prime submodule of U has a non-zero intersection with V (2). A submodule V of U is called closed if V has no proper essential extensions inside U (1, P.18). Ahmed and Abbas introduced the concept of Stclosed submodule, where a submodule V of U is said to be St-closed, if V has no proper semiessential extensions inside U (3).
In this paper, we introduce and study a new class named St-polyform modules. This type of modules is contained properly in some classes of modules such as polyform, strongly essentially quasi-Dedekind and -non St-singular modules. An R-module U is called polyform if for every submodule V of U and for any homomorphism : VU, ker is closed submodule in U (4). A module U is called strongly quasi-Dedekind, if Hom R ( We define in this work a proper class ofnonsingular modules named -non St-singular. We define St-polyform as follows: an R-module U is called St-polyform, if for every submodule V of U and for every homomorphism : VU, ker is Stclosed submodule in V. We verify that an Stpolyform module is smaller than all of the classes: polyform, strongly quasi-Dedekind, -nonsingular and -non St-singular modules, see remark 2, proposition 30, proposition 40 and proposition 56. Beside that we give another generalization for Stpolyform modules. This work consists of three sections. In the first section we provide another characterization of Stpolyform modules, we show that a module U is Stpolyform if and only if for each non-zero submodule V of U and for each non-zero homomorphism : VU; ker is not semi-essential submodule of V, see theorem 4. Also we present the main properties of St-polyform module, for example we show in proposition 7 the existence of St-polyform in certain class of modules, also we prove in the proposition 11 where U 1 and U 2 are R-submodules of U. Let V 1 be a non-zero submodule of U 1 , and : V 1  U 1 be a non-zero homomorphism. Consider the following sequence: The converse of proposition 6 is not true in general; for example each of Z 10 and Z 6 are Stpolyform Z-modules; see 3vii, but Z 10 ⨁Z 6 is not St-polyform Z-module. Recall that an R-module U is called Artinian if every descending chain of submodules in U is stationary (1,P.7). The following proposition indicates the existence of St-polyform submodules in certain class of modules. Proposition 7: Every nonzero Artinian module has a submodule which is an St-polyform. Proof: Let U be a non-zero Artinian module, and V be a submodule of U. If V is St-polyform, then we are done. Otherwise there exists a submodule V 1 of V and a homomorphism 1 : V 1 V with ker( 1 ) ≰ St 1 and ker( 1 ) ≤ St V 2 for some proper submodule V 2 of V 1 . Now, if V 1 is St-polyform, then we are through, otherwise there exists a submodule V 3 of V 2 and a homomorphism 2 : V 3 V 2 with ker( 2 ) ≰ St 3 and ker( 2 ) ≤ 4 for some proper submodule V 4 of V 3 . We continue in this manner until we arrive in a finite number of steps at a submodule which is an St-polyform submodule. Otherwise, we have an infinite descending chain VV 1 V 2  . . . . . of submodules of the module U. But this is a contradiction, since U is Artinian. Therefore U contains an St-polyform submodule. ∎ Proposition 8: Let U be an R-module. If either V 1 or V 2 are St-polyform module, then V 1 ∩ V 2 is Stpolyform module. Proof: Assume that V 1 is St-polyform module. Let V be a non-zero submodule of V 1 ∩ V 2 , and let f: VV 1 ∩V 2 be a non-zero homomorphism. Consider the following sequence: Recall that an R-module U is called scalar if for any  End R (U), there exists rR such that (x)=rx xU, where End R (U) is the endomorphism ring of U (5). Proof: Since U is finitely generated and multiplication, then U is a scalar module (7), and the result follows by proposition 9. ∎ Proposition 11: Let U be an R-module. If W≤ sem V for every submodule V of U, such that Hom R ( V W , U) = 0, then U is a St-polyform module. Proof: Assume U is not St-polyform module, so there exists a submodule V of U and a non-zero homomorphism : V U such that ker ≤ sem U. Define : . We can easily show that is well defined and homomorphism. Since is a non-zero homomorphism, then is also non-zero, thus Hom R ( V W , U)≠0. But this contradicts our assumption, therefore ker ≰ sem U. Recall that an R-module U is called injective if for every monomorphism : AB where A and B be any R-modules, and for every homomorphism : AU, there exists a homomorphism h: B U such that ℎ ∘ = (8, P.33). A module U is called quasi-injective if it is U-injective R-module (8, P.83). The injective hull (quasi-injective hull) of a module U is defined as an injective (quasiinjective) module with essential extension of U, it is denoted by E(U) (respectively U ̅ ) (8, P.39). Clark and Wisbauer in (9) proved that a module U is polyform if its quasi-injective hull is polyform. As analogue of that, we have the following result. Proposition 13: Let U be an R-module. If the injective hull E(U) of U is St-polyform module, then U is St-polyform module. Proof: Let V be a non-zero submodule of U, and :VU be a non-zero homomorphism. Suppose the converse is not true, that is ker ≤ sem V. Consider the following Fig. 1. , and by our assumption ker ≤ sem V, then by transitivity of semi-essential submodules ker ≤ sem E(V) (2). On the other hand, clearly ker  ker , therefore ker ≤ sem E(V) (2), which is a contradiction. Therefore, ker ≰ sem V, i.e V is an St-polyform module. ∎ In example 3ix, we verified that a submodule of St-polyform may not be St-polyform. In the following proposition, we satisfy that under certain condition.

St-polyform and Polyform modules:
In this section, we investigate the relationships of St-polyform module with polyform and small polyform modules. Besides that, we introduce another generalization for St-polyform modules.
In the previous section, we verified that the class of St-polyform modules is a proper subclass of polyform modules. In the following theorems, we use certain conditions under which St-polyform module can be polyform module. Before that; an Rmodule U is called fully prime if every proper submodule of U is prime (2). Theorem 19: Let U be a fully prime R-module, then U is St-polyform if and only if U is a polyform module. Proof: ⇒) By remark 2. ⇐) Assume that U is polyform module, and let V be a submodule of U, and : V U be a homomorphism. Since U is polyform, then ker is closed submodule in U. But U is fully prime, then ker is an St- Recall that an R-module U is called fully essential, if every semi-essential submodule of U is essential (2). Theorem 20: Let U be a fully essential R-module, then U is St-polyform if and only if U is a polyform module. Proof: ⇒) By remark 2. ⇐) Let V be a non-zero submodule of U, and : V U be a non-zero homomorphism. Since U is polyform, then ker ≰ e V. But U is fully essential; therefore, ker ≰ sem V (2), that is U is St-polyform module. ∎ The following proposition shows that the class of St-polyform domain coincides with the class of polyform domain. Proof: ⇒) It is straightforward. ⇐) Assume that U is essentially St-polyform, and let V be a non-zero submodule of U, and : V U be a non-zero homomorphism. Since U is a uniform module so V ≤ e U. But U is essentially St-polyform; therefore, ker ≰ sem V; that is U is an St-polyform module. ∎ By replacing uniform module by hollow and essential submodule by small, we have the following; and the proof is in a similar way. Proposition 27: Let U be a hollow module, then U is St-polyform if and only if U is small St-polyform. We can summarize the main results of this section by the following implications of modules:

St-polyform and other related concepts:
This section is devoted to study the relationships of St-polyform with some related concepts such as quasi-Dedekind and some of its generalizations, -nonsingular, injective, extending, Baer and -non St-singular module.
Recall that an R-module U is called quasi-Dedekind, if for every non-zero homomorphism End(U), ker =0 (11). Remark 28: It is worth mentioning that Stpolyform modules and quasi-Dedekind modules are independent; for example the Z-module Z 6 is Stpolyform module see example 3vii, but not quasi-Dedekind. On the other hand, Z is quasi-Dedekind (11), but not St-polyform, see example 5ii. Proposition 29: Let U be a semi-uniform module. If U is St-polyform then U is a quasi-Dedekind module. Proof: Assume that U is St-polyform module, and let End(U). If V be a non-zero submodule of U, then we have the following sequence: V → U→ U Where is the inclusion homomorphism. Suppose that ker ≠ 0, since U is St-polyform. Note that ∘ ≠ 0. Since U is St-polyform, then ker( ∘ ) ≰ sem V, hence ker( ∘ ) ≰ sem U (2). But this is a contradiction since U is a semi-uniform module, thus ker = 0. ∎ The converse of proposition 29 is not true in general, for example Z 2 is a quasi-Dedekind module, but not St-polyform. Recall that an R-module U is called strongly essentially quasi-Dedekind if for each non-zero homomorphism End R (U), ker ≰ sem U (5). Proposition 30: Every St-polyform module is strongly essentially quasi-Dedekind. Proof: Let U be St-polyform module. Let V be a non-zero submodule of U, and : VU be a nonzero homomorphism. By assumption ker is not semi-essential submodule in V. In particular, all non-zero endomorphisms of U have kernels which are not semi-essential in U, proving our assertion. ∎ The converse of proposition 30 is not true in general, for example Z 2 is strongly essentially quasi-Dedekind module (5, Ex (1.11)) but not Stpolyform as we saw in remark 2. In the following theorem we use a condition under which the converse is true. Proposition 31: Let U be a quasi-injective Rmodule then U is U is St-polyform if and only if U is a strongly essentially quasi-Dedekind module. Proof: ⇒) By proposition 30. ⇐) Let V be a non-zero submodule of U, and :VU be a non-zero homomorphism. Consider the following Fig. 2.

Figure 2. The diagram of injective module U
where : V U is the inclusion homomorphism. Since U is quasi-injective, then there exists a homomorphism : UU such that ∘ = . Now, End(U) and U is essentially quasi-Dedekind; therefore, ker ≰ sem U. But ker  ker , then by transitivity of semi-essential submodule, ker ≰ sem U (2), and we are done. ∎ In (3) Ahmed and Abbas proved that if every submodule of U is St-closed, then every submodule of U is direct summand. This motivates us to introduce the following. Definition 32: An R module U is called Stsemisimple if every submodule of U is St-closed. This concept is clearly a proper subclass of semisimple modules, and we can easily prove the following. We think that the converse of the remark 33 is not true in general, but we cannot find example. Definition 34: Let U be an R-module. We define St-singular submodule as follows: {uU| ann R (u) ≤ sem R} It is denoted by St- Recall that an R-module U is callednonsingular, if for each End R (U); ker ≤ e U, then =0 (6, P.95). In other words, for every nonzero homomorphism End R (U); ker ≰ e U. As example for this class of modules is Z-module Z p , it is -nonsingular for every prime number P, since Z p is a simple module; therefore, all non-zero endomorphisms are automorphisms. Remark 39: The concept of -nonsingularity is strictly weaker than the concept of nonsingularity for modules (6, Ex(4.1.10), P.96), where an Rmodule U is called nonsingular if Z(U)=0, where Z(U)= {uU| ann R (u)≤ e R} (1, P.30). Proposition 40: Every St-polyform module isnonsingular. Proof: Let U be St-polyform module. Let V be a non-zero submodule of U, and : V U be a nonzero homomorphism. By assumption, ker ≰ sem V. As we take V=U, then we obtain : U  U, and ker ≰ sem U. Since every essential submodule is semiessential (2), then ker ≰ e U, hence U isnonsingular. ∎ The converse of proposition 40 is not true in general as the following examples show: It is clear that is a non-zero homomorphism, then ker ={(x,0)V| (x,0)=(0,0 ̅ )}=2Z⨁(0). We can easily verify that 2Z⨁(0) ≤ sem V, hence U is not St-polyform module. On the other hand, U is -nonsingular Z-module (12). The following proposition gives a partial equivalence between St-polyform and nonsingular modules. Theorem 42: Let U be a fully essential quasiinjective module, then U is St-polyform if and only if U is -nonsingular provided that Hom R (V,U) ≠ 0. Proof: ⇒) By proposition 40. ⇐ ) Suppose that U is a -nonsingular, and let V be a non-zero proper submodule of U. Let : V U, Since Hom R (V,U)≠0, so we can take ≠0. Consider the following Fig. 3. where : V U is the inclusion homomorphism. Since U is quasi-injective, then there exists a homomorphism : UU such that ∘ = . Now,  End R (U) and U is -nonsingular, thus ker ≰ e U. But ker  ker , thus ker ≰ e U. Since U is fully prime, then ker ≰ sem U. ∎ Corollary 43: Let U be a fully prime injective module. Then U is an St-polyform module if and only if U is -nonsingular. Proof: Since every fully prime module is fully essential (2), and End R (U) ≠ 0, then the result follows by theorem 42. ∎ Lemma 44: (11) If U is an injective module, then J(End R (U)) = {  End R (U)| ker ≤ e U}. Corollary 45: Let U be a fully essential module. If U is injective and J(End R (U)) = 0, then U is Stpolyform.
In order to verify the relation of St-polyform with Baer module, we need to introduce the following proposition. Proposition 50: Every Baer quasi-injective module is polyform. Proof: Let V be a non-zero submodule of U, and : VU be a non-zero homomorphism. Suppose the converse; that is ker ≤ e V. Consider the following Fig. 4: where : V U is the inclusion homomorphism. Since U is quasi-injective, then there exists a homomorphism : UU such that ∘ = . Now, End R (U) and U is Baer, so ker = ann S = e, e 2 =e, and S= End R (U). This implies that ker is direct summand of U (12). Since ker( ∘ )  ker , then clearly ker( ∘ ) is a direct summand of V. But ∘ = , thus ker is direct summand of V. On the other hand, ker ≤ e V, therefore ker =V, hence = 0 which is a contradiction with assumption, thus ker ≰ e V. ∎ Corollary 51: For a fully prime (or fully essential) module, every Baer quasi-injective module is Stpolyform. Proof: Since in the class of fully prime (or fully essential) modules the concept of essential submodules coincides with the concept of semiessential, so the proof is in similar of the proposition 50. ∎ Proposition 52: Let U be an extending module. If U is St-polyform, then U is a Baer module. Proof: Since U is St-polyform, then by proposition 40, U is -nonsingular. On the other hand, U is extending, so U is Baer (6, Lemma(4.1.17), P.97). ∎ Theorem 53: Let U be an quasi-injective module. Consider the following statements: