T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules

A new generalizations of coretractable modules are introduced where a module M is called t-essentially (weakly t-essentially) coretractable if for all proper submodule K of M, there exists f∈End(M), f(K)=0 and Imf≤tes M (Im f +K ≤tes M). Some basic properties are studied and many relationships between these classes and other related one are presented.

In (6), Hadi and Al-Aeashi introduced two classes related coretractable modules which are essentially coretractable and weakly essentially coretractable modules, where each of these classes is contained in the class of coretractable modules. "A module ℳ is called essentially coretractable if for each proper submodule of ℳ, there exists 0≠f: ℳ/ → ℳ such that Imf ≤ ess ℳ and ℳ is weakly essentially coretractable if Imf+ ≤ ess ℳ" (6). In § 2 The notion t-essentially coretractable was studied, a module ℳ is called t-essentially coretractable if for each proper submodule of ℳ, there exists 0≠f: ℳ/ → ℳ such that Imf ≤ tes ℳ. Also give some connections between it and other related classes of modules. In § 3 , the notion weakly t-essentially coretractable modules are introduced and studied, as a generalization of weakly essentially coretractable module, ℳ is called weakly t-essentially coretractable if for each proper submodule of ℳ, there exists 0≠f:ℳ/ → ℳ such that f(ℳ/ )+ ≤ tes ℳ. Many other connections between these classes and other related are given. Recall that " a module ℳ is called epicoretractable if for each proper submodule of ℳ, there exists an epimomorphism f∈Hom(ℳ/ ,ℳ)" (7). "A module ℳ is called hopfian if each epimomorphsim f∈End(ℳ), then f is monomorphism. And ℳ is antihopfian module if ℳ/ ≅ ℳ for all proper submodule of ℳ" (8), Clearly any antihopfian module is epi-coretractable. "An R-module M is called quasi-Dedekind if for each proper submodule of ℳ, Hom(ℳ/ , ℳ)=0 (9)" and " ℳ is coquasi-Dedekind module if for each 0≠ f∈ End(ℳ), f is an epimorphism" (10). "A module ℳ is called C-coretractable (Ycoretractable) if for all proper closed (y-closed) submodule of ℳ, there exists 0≠f:ℳ/ → ℳ" (11), (7), where a submodule N of ℳ is called yclosed if ℳ/N is nonsingular module" (4). Note that every y-closed submodule of ℳ is closed but the converse may not be true. They are equivalent if ℳ is nonsingular (4).
(3) The two concepts t-essentially coretractable module and semisimple are independent, see the following examples: The Z-module Z 4 is tessentially coretractable since it is essentially coretractable by (6, Example (2.2(4)), but Z 4 is not semisimple. The Z 6 -module Z 6 is semisimple but it is not tessentially coretractable see Rem. & Exa. (2(3)). Also ℳ=Z 2 ⨁Z 2 as Z 2 -module is semisimple module but it is not t-essentially coretractable (4) Clearly every antihopfian module is t-essentially coretractable (since every antihopfian is essentially coretractable (6) which implies tessentially coretractable). (5) Every t-essentially coretractable module is Ccoretractable, Y-coretractable module. The converse is not true, see Z as Z-module is C-coretractable and it is not t-essentially coretractable. (6) A module ℳ is t-essentially coretractable if and only if ℳ is t-essentially coretractable ℛ ̅module (ℛ ̅ = ℛ/ann ℳ). (7) If ℛ is t-essentially coretractable ring, ℳ is faithful cyclic ℛ-module. Then ℳ is tessentially coretractable (8) A ring ℛ is t-essentially coretractable if and only if for each proper ideal I of ℛ, there exists r∈ ℛ, r≠0 such that r ∈annI and <r>≤ tes ℛ. Proposition 1: Let ℳ be a nonsingular module. Then ℳ is essentially coretractable if and only if it is t-essentially coretractable. Proof: (⇒) It is obvious since every essential submodule is t-essential..
It is to be noted that a direct summand of coretractable module need not be coretractable (12). Also it is to be noted that any direct summand of essentially coretractable module is essentially coretractable see (6, corollary(2.7)), however if ℳ=C⊕N is a Z 2 -torsion R-module and C is a cogenerator (where an R-module ℳ is called a cogenerator if for any R-module N and 0≠x∈N, there exists f: N→ ℳ such that f(x)≠0. (12)) and N is any R-module, then ℳ is coretractable module by (12, proposition(1.5)) and so by proposition (6) M is t-essentially coretractable but N need not be coretractable and hence N need not be t-essentially coretractable. Recall that "A module ℳ is compressible if it can be embedded in any nonzero its submodule"(13). Proposition 6: Let ℳ be a compressible tessentially coretractable module and D be a direct summand of ℳ, then D is t-essentially coretractable. Proof: Since D≤ ⊕ ℳ, so ℳ=D⊕W for some W≤ ℳ. Let K< D, hence K⊕W≤ ℳ and so ∃0≠ f: ℳ ⊕W →ℳ with Imf≤ tes ℳ. Now since ℳ ⊕W ≅ and ℳ is compressible module, so ∃g: ℳ→D, g is monomorphism. Then g•f: →D and g•f( )= g( f( ) ). But Imf ≤ tes ℳ and g is monomorphism, so g( f( )) ≤ tes D by (14, proposition(1.1.23)). Also is t-essentially coretractable.
Proof: Since 2 (ℳ) is y-closed, then the required condition hold by Proposition (7). Corollary 3: Let ℳ=D⊕W be a t-essentially coretractable module such that 2 (ℳ) ⊆W. Then D is t-essentially coretractable module. Proof: Let ℳ=D⊕W . Since W is direct summand, so W is closed, but 2 (ℳ) ⊆W by hypothesis and hence W is y-closed submodule. Hence by Proposition (7), Note that" a finite direct sum of coretractable modules is coretractable module", see(2, Proposition(2.6)).

Remark 2:
The direct sum of t-essentially coretractable modules need not be t-essentially coretractable module, for example: Let ℳ=Z 2 ⊕Z 2 as Z 2 -module. ℳ is not t-essentially coretractable but Z 2 is t-essentially coretractable Z 2 -module.

Conclusion:
In this paper, the notions of t-essentially and weakly t-essentially coretractable modules are defined as a generalization of essentially and weakly essentially coretractable module. Also, several results are given. Further the completely weakly t-essentially coretractable rings are defined and investigated.