Essentially Second Modules

In this paper, as generalization of second modules we introduce type of modules namely (essentially second modules). A comprehensive study of this class of modules is given, also many results concerned with this type and other related modules presented.


Introduction
In this research all rings are associative with identity and all modules are unitary right modules. For a right -module we write .Agayev in [1] defined and studied r-semisimple modules, where an -module is said to be r-semisimple if for any right ideal of , is a Direct summand of (briefly The class of . r-semisimple modules contains the class of semisimple modules , also contains the class of second modules, where an -module is named second if and for each , either or [2]. Equivalently is second if for each ideal of , either or [2]. Annine in [3], [4] introduced the class of coprime modules.  An -module is coprime if ( ( for each proper submodule of ( , where ( . Wijayanti in [5] called an -module is coprime if ( ( for each fully invariant submodule of , where a submodule of is called fully invariant if for each endomorphism ( ( ), (  [6]. However, coprime module (in sense of Annine), coprime modules (in sense of Wijayanti) and second modules are coinciding.
Iraqi Journal of Science, 2019, Vol. 60, No. 6, pp: 1374-1380 1375 In this paper, we give another generalization of second modules. An -module is an essentially second (shortly ess. second) if for each ideal of , either or . where a submodule of is essential (briefly ) if whenever ( , then ( [7]. Equivalently if and only if for each ,  ; [7]. It is clear that every second and uniform modules are ess. second but the converses are not true, see Remarks 2.2(2), (3).
In section two, we give the basic properties of ess. second modules such as in the class of multiplication modules, ess. second modules and uniform modules are equivalent (see, Corollary 2.4) . Every pure submodule (hence every direct summand) of ess. second modules is an ess. second module (Proposition 2.12), but the direct sum of ess. second modules may be not ess. second (see Remark 2.8). Also, if is an ess. second and is a closed submodule, then is an ess. second module (see Proposition 2.9).
In section three we present many relationships between ess. second modules and other related concept such as prime modules, r-semisimple modules (see Proposition 3.1, Theorem 3.2 and Proposition 3.3).

Essentially second modules
if M is an -module, a submodule of is second submodule if for each ideal of , either ( or [2]. A module is second if it is a second submodule of . A ring is a second if is a second -module. We define:  [11]. Equivalently is coquasi-Dedekind if for each ( is an epimorophrism. We present the following Definition 3.6: An -module is to be essentially coquasi-Dedekind if for each ( . Note that Sahra in [11] gave the following: an -module is called essentially coquasi-Dedekind if for each (0) ( ( . However our definition is different of that was given in [11]. Examples 3.7: 1-Every simple module (and the -modules are ess. coquasi-Dedekind in sense of Definition 3.6, but it is not ess. coquasi-Dedekind in sense of [11]. 2-Consider as -modules, is an ess. couasi-Dedekind in sense of [11]. But it is not ess.

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is an ess. second left -module. As we mention in the introduction the second module is called coprime by some authors, see [2,13]. Sahera in [11] introduced the concept ess. coprime as a generalization of coprime ( second module) where an -module is referred by an ess. coprime if for each , either or ( , where ( . Notice that the concept ess. second is independent with ess.coprime [11] It is known that for every second -module ( a prime ideal. of . However this is not true for ess. second module as we have:-the -module is an ess. Second (since it is uniform ) and ( which is not a prime ideal. of . In [13] we define the concept essential prime (briefly ess . prime ) as follows : an -module M is said to be an ess. prime whenever ( ( for all .  We state and prove the pursue :  (3) Since is prime., then is an ess. prime. But is prime implies is second by (part (2)  (1)), hence is ess. second.
It is known that if is an Artinian ring or a Boolean ring, then every prime ideal. is maximal. Hence we get. Corollary 3.14: Let be an -module where is an Artinian ring or Boolean ring. Then the pursue is synonymous.  [14]. In the last part of Lemma 1.1 in [14]. If is a module over a commutative such that every prime ideal . is maximal, then is second iff is a homogenous semisimple. Corollary 3.16: If is an -module, where is a commutative ring. such that every prime ideal. is maximal (hence if is Artinian ring or Boolean or Von Neumman regular). Then the pursue are synonymous: 1-is second ; 2-is prime.;

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is an ess. prime and ess. second module;

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is a homogenous semisimple. Proposition 3.17: Let be multiplication module over a ring . Then is a second if and only if is a homogenous semisimple. Proof:  Since is a multiplication module then for each proper submodule N of M, N=M [N:M ].=M ann . Because M is second, ann =ann M , hence N=M ann M=0 Then is simple . Thus is homogenous semisimple.  It is given in [14]. Corollary3.18: Let be a commutative ring . Then is second if and only if is homogenous semisimple