H-essential Submodules and Homessential Modules

The main goal of this paper is introducing and studying a new concept, which is named H-essential submodules, and we use it to construct another concept called Homessential modules. Several fundamental properties of these concepts are investigated, and other characterizations for each one of them is given. Moreover, many relationships of Homessential modules with other related concepts are studied such as Quasi-Dedekind, Uniform, Prime and Extending modules.


Introduction
Throughout this paper, all rings are commutative with identity and all modules are unitary left Rmodules. "A submodule V of a module U is called essential (simply V ≤ e U), if the intersection of V with any non-zero submodule of U is not equal to zero" [1]. In this paper, we introduce a new concept, named Homessential module. This concept needs to define a certain type of submodules named Hessential submodules, where a proper submodule V of U is called H-essential, if for each non-zero homomorphism Hom R ( ); ( ) is an essential submodule of U. A module U is called Homessential if every proper submodule of U is H-essential.
In section two; we investigate the main properties of H-essential submodules; we determine the trace of the quotient module over any H-essential submodule, see Proposition (2.8). Also, we show that the intersection of any two H-essential submodules is also H-essential, see Proposition (2.11). Furthermore, another characterization of H-essential submodule is given, see Theorem (2.12), and we show that every rational submodule is H-essential (see 2.13). Moreover, we prove that in the class of polyform module, every essential submodule is H-essential, Corollary (2.15).

ISSN: 0067-2904
Section three of this paper is devoted to introduce and study the concept of Homessential module, we give another characterization of Homessential module, see Proposition (3.6). Also, the relationships of Homessential module with some other related concepts such as quasi-Dedekind, uniform and prime and extending modules are studied, see the results (3.9), (3.10), (3.12), (3.14) (3.15), (3.16) and (3.17).

H-essential Submodules
This section is devoted to studying the main properties of H-essential submodules. Definition (2.1): Let U be an R-module. A proper submodule V of U is called H-essential, if for each non-zero homomorphism Hom R ( ); ( ) is essential submodule of U. Examples (2.2): 1. Consider the -module 4 , the submodule ( ̅ ) of Z 4 is H-essential, since the non-zero homomorphism in ( ̅ ) is only the inclusion homomorphism. In fact ( ̅ )=( ̅ ), and ( ̅ ) is essential submodule of 4 .

2.
In the -module 6 , the submodule ( ̅ ) is not H-essential. In fact; there exists a homomorphism f: is not essential submodule of 6 .

Proposition (2.3):
If V is H-essential submodule of U, then rU ≤ e U for each (0≠) r ann R (V "Recall that a ring R is called Noetherian if every ideal of R is finitely generated" [2, P.55]. Proposition (2.7): Let R be a Noetherian ring, and U be an R-module. If V is H-essential submodule of U, then ann R (V)U ≤ e U. Proof: Since R is a Noetherian ring, then ann R (V ) is finitely generated ideal of R. That is ann R (V) = (r 1 , r 2 ,…, r n ) for some r i ann R (V ). This implies that ann R (V )U = ∑ U . By Proposition (2.3), r i U ≤ e U for each i=1,2, . . ., n, and by Lemma (2.6), (ann R (V ))U ≤ e U.  "For R-modules U and V, the trace of an R-module U is defined by T V (U) =∑ (U) where Hom R (U, V ) ([2], p.27)". If V=R, then the trace of U on R is denoted by T(U). Proposition (2.8): Let U be an R-module and T(U) is the trace of U on R. If V is an H-essential submodule, then either T( ) is zero or T( )U is essential submodule of U.

Proof: Suppose that T( ) ≠ 0 and (0≠)
Hom R ( , R). For each u U, one can define u : R → U by u (r) = ru; for all r R. It is clear that u is well defined and homomorphism. So ( u ) Hom R ( , U). Since V is H-essential submodule of U, then (f u ) ≤ e U for each u U. This implies that 2. Consider the -module . is H-essential submodule of (as we will see in Example 2.14), but ( )= since ( ) = 0. Proposition (2.11): Let A and B be H-essential submodules of an R-module U, then A ∩ B is an Hessential submodule. Proof: If A∩B = 0, then we are done. Assume that A∩B ≠ 0 and 0 ≠ Hom R ( , U). It is clear that Hom R ( , U) ≤ Hom R ( ; U) + Hom R ( , U) [3]. So for all Hom R ( , U), = + where Hom R ( , U) and Hom R ( , U) with ( ) ≤ e U and ( ) ≤ e U implies ( ) + ( ) ≤ e U. Hence ( ) ≤ e U, hence we are done.  Theorem (2.12): Let U be an R-module, and 0 ≠ V ≤ U. Then the following statements are equivalent:  Example (2.14): Consider the -module , where is the set of all rational numbers. Since is rational submodule of hence by Proposition (2.13), is H-essential submodule of .
"Recall that an R-module U is called polyform if every essential submodule is rational" [4]. By proposition (2.13), we conclude the following. Corollary (2.15): If U is polyform module, then every essential submodule of U is H-essential.

Homessential Module
In this section, we introduce the class of Homessential module. Definition (3.1): An R-module U is called Homessential module, if every proper submodule of U is H-essential. A ring R is called Homessential, if R is Homessential R-module. Examples (3.2): 1. The zero R-module is a Homessential module, since (0) has no non-zero H-essential submodule V of (0) such that (V) is not essential submodule of (0).

2.
A semisimple module is not Homessential module. 3. Every integral domain R is Homessential R-module. 4. Every uniform module is Homessential module, where an R-module U is called uniform if every non-zero submodule is essential in U [2]. 5. Every monoform module is Homessential, where a module U is called monoform if every nonzero submodule is rational [5], in fact this follows directly from Proposition (2.13).
Proof: 3. Let R be an integral domain, and I be a non-zero ideal of R. We can easily show that in an integral domain every non-zero ideal is an essential ideal. This implies that is singular R-module, hence, it is torsion. On the other hand, R is torsion free R-module, therefore Hom R ( ) = 0, that is R is Homessential module. In particular, Z is Homessential. Proposition (3.4): A direct summand of a Homessential module is Homessential. Proof: Let U=U 1 U 2 be a Homessential module, where U 1 and U 2 be R-submodules of U, and V be a submodule of U 1 . Let 0 ≠  End R (U 1 ) such that V ≤ ker . Consider the following sequence of homomorphism: The following theorem gives another characterization of Homessential module. Proposition (3.6): Let U be an R-module, and V be a non-zero submodule of U. Then the following statements are equivalent: 1. U is a Homessential module.

For each
with ker ≠0; . Proof: (1) (2) Let U be a Homessential module, and End R (U), with ker ≠ 0. Put V = ker , by assumption V is H-essential submodule of U. By Theorem (2.12), ( ) . (2) (1) Let V be a non-zero proper submodule of U, and Hom R ( ). Note that V ≤ ker( ), where : U is the natural epimorphism. Since V≠ 0, then ker( ) ≠ 0. By Theorem (2.12), V is an H-essential submodule, hence U is Homessential module  "Recall that a submodule V of a module U is said to be closed if V has no proper essential submodule in U" [1, p.18] Proposition (3.7): Let U be a Homessential module. If U is a semisimple module, then for every Hom R ( ), f is an epimorphism.
Proof: Let V is a proper submodule of U and Hom R ( ). By the definition of Homessential module, ( ) . But U is semisimple, so that ( ) is closed submodule of [7]. This implies that ( ) = U, hence is an epimorphism.  "Recall that an R-module U is said to be quasi-Dedekind, if for every non-zero homomorphism End(U), ker =0" [8] Proposition ( Corollary (3.9): If U is a finitely generated quasi-Dedekind module, then U is Homessential. Proof: Since U is a finitely generated quasi-Dedekind module, then U is uniform [8], and by Remark (3.3)(4), U is Homessential.  "A ring R is called regular (in the sense of Von Newmann) if for each aR there is exists bR such that a=aba" [2, P.4]. Under certain condition Homessential module can be quasi-Dedekind as the following proposition shows. Proposition (3.10): Let U be a Homessential R-module. If is a regular ring, then U is quasi-Dedekind. Proof: Let 0≠  . We have to show that ker =0, otherwise; since U is a Homessential module, so by Proposition (3.6), ( ) . But , then U= (U) [7, Exc. 17(a), P.272]. This implies that . Since ker ≠0, so is not essential in U, which is a contradiction.  Remark (3.11): The condition " . is regular" cannot be dropped from Proposition (3.10), for example; Z 4 is Homessential Z-module but not quasi Dedekind, since . = which is not regular.
It is known that if U is a semisimple module, then is regular [9, Cor.(2.22, P.52)]. So by this fact and Proposition (3.10) we have the following. Corollary (3.12): If U is semisimple and Homessential module, then U quasi-Dedekind. Remarks (3.13): 1. The condition "semisimple" in Corollary (3.12) cannot be dropped. For example, the -module is uniform module, so it is Homessential (see Remark (3.3)(4)) but it is not quasi-Dedekind module, since it is not semisimple module.

2.
In general, Homessential module need not be prime module; for example: the -module 4 is Homessential but not prime module.
The following Corollary gives a necessary condition under which Homessential module can be prime. Corollary (3.14): Let U be a Homessential module. If U is semisimple, then: 1. U is a prime module. "Recall that an R-module P is said to be projective if for every epimorphism f: B C and for every homomorphism g: P C there is a homomorphism h: P B with g = hf" [7, Def. (5.3.1) (b), p. 116]. Corollary (3.15): Let R be a regular ring and U is a finitely generated projective module. If U is Homessential, then U is a quasi-Dedekind module. Proof: Assume that U is a Homessential module. Since U is a finitely generated projective module over regular ring, then is regular [7,Exc.17(c), P.272], and by Proposition (3.10), we are done.  "Recall that an R-module U is called Z-regular, if every cyclic submodule of U is direct summand and projective" [10]. Corollary (3.16): Let U be a Z-regular module. If U is a Homessential module, then U is quasi-Dedekind. Proof: Since U is a Z-regular module, then is regular [10]. But U is Homessential, then by proposition (3.10), U is a quasi-Dedekind.  "Recall that an R-module U is called extending if every submodule of U is essential in a direct summand of U" [2, p.118], and U is indecomposable, if the only decomposition U=A B are those in which either A=(0) or B=(0) [7, p.285]. Proposition (3.17): Every non-zero extending and indecomposable module is Homessential. Proof: Since every non-zero extending and indecomposable module is uniform, then the result follows by Remark (3.3)(4). 