Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986

DOI: http://dx.doi.org/10.21123/bsj.2021.18.1.0156

T-Essentially Coretractable and Weakly T-Essentially Coretractable Modules

Inaam Mohammed Ali Hadi*1 Shukur Neamah Al-Aeashi 2 Farhan Dakhil Shyaa3

1 Department of Mathematics, College of Education for Pure Science/ Ibn Al-Haitham, University of Baghdad, Iraq. 2 Department of Mathematics, College of Education, University of Kufa, Kufa, Iraq. 3 Department of Mathematics, College of Education, University of Al-Qadisiyah, Qadisiyah, Iraq. *Corresponding author: [email protected] *,[email protected], [email protected] *ORCID ID: https://orcid.org/0000-0002-1058-7410, https://orcid.org/0000-0002-9886-3247, https://orcid.org/0000- 0003-1123-1974

Received 24/2/2019, Accepted 25/4/2020, Published Online First 6/12/2020, Published 1/3/2021

This work is licensed under a Creative Commons Attribution 4.0 International License.

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

Key words: Coretractable module, Essentially coretractable module, T-essentially coretractable module, Weakly t-essentially coretractable module, Weakly essentially coretractable module.

Introduction: In this work, all rings have identity and all weakly essentially coretractable if Imf+퐾 ≤ess ℳ" modules are unitary left R-modules. A coretractable (6). In §2 The notion t-essentially coretractable was module appeared in(1). However, Amini(2), studied studied, a module ℳ is called t-essentially this class of modules, where "ℳ is called coretractable if for each proper submodule 퐾 of ℳ, coretractable if for all proper submodule 퐾 of ℳ, there exists 0≠f: ℳ/퐾→ ℳ such that Imf ≤tes ℳ. there exists 0≠f∈Hom(ℳ/퐾, ℳ)" (1). Next, Hadi Also give some connections between it and other and Al-Aeashi defined strongly coretractable related classes of modules. In§3, the notion weakly module where" a module ℳ is called strongly t-essentially coretractable modules are introduced coretractable if for each a proper submodule 퐾 of and studied, as a generalization of weakly ℳ, there exists a nonzero homomorphism essentially coretractable module, ℳ is called f: ℳ/퐾→ ℳ such that Imf+퐾=ℳ"(3). "A weakly t-essentially coretractable if for each proper submodule 퐾 of ℳ is called essential in ℳ submodule 퐾 of ℳ, there exists 0≠f:ℳ/퐾→ ℳ (퐾 ≤ess ℳ), if 퐾 ∩S=(0), S≤ ℳ implies S=(0)" (4), such that f(ℳ/퐾)+퐾 ≤tes ℳ. Many other and 퐾 is t-essential submodule (퐾 ≤tesℳ) if for connections between these classes and other related every submodule S of ℳ, S∩ 퐾 ⊆ Z2(ℳ) implies are given. Recall that " a module ℳ is called epi- that S⊆Z2(ℳ), where Z2(ℳ) is called the second coretractable if for each proper submodule 퐾 of ℳ, and is defined by there exists an epimomorphism f∈Hom(ℳ/퐾,ℳ)" ℳ 푍 (ℳ) Z( )= 2 . Clear that every essential submodule (7). "A module ℳ is called hopfian if each 푍(ℳ) 푍(ℳ) epimomorphsim f∈End(ℳ), then f is is t-essential, but not conversely. However they are monomorphism. And ℳ is antihopfian module if coincide in the class of nonsingular module (5)". ℳ/퐾 ≅ ℳ for all proper submodule 퐾 of ℳ" (8), In(6), Hadi and Al-Aeashi introduced two Clearly any antihopfian module is epi-coretractable. classes related coretractable modules which are "An R-module M is called quasi-Dedekind if for essentially coretractable and weakly essentially each proper submodule 퐾 of ℳ, Hom(ℳ/퐾, ℳ)=0 coretractable modules, where each of these classes (9)" and " ℳ is coquasi-Dedekind module if for is contained in the class of coretractable modules. each 0≠ f∈ End(ℳ), f is an epimorphism"(10). "A "A module ℳ is called essentially coretractable if module ℳ is called C-coretractable (Y- for each proper submodule 퐾 of ℳ, there exists coretractable) if for all proper closed (y-closed) 0≠f: ℳ/퐾→ ℳ such that Imf ≤ess ℳ and ℳ is

156 Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986 submodule 퐾 of ℳ, there exists 0≠f:ℳ/퐾→ ℳ" coretractable and it is not t-essentially (11), (7), where a submodule N of ℳ is called y- coretractable. closed if ℳ/N is nonsingular module"(4). Note that (6) A module ℳ is t-essentially coretractable if and every y-closed submodule of ℳ is closed but the only if ℳ is t-essentially coretractable ℛ̅- converse may not be true. They are equivalent if ℳ module (ℛ̅= ℛ/ann ℳ). is nonsingular (4). (7) If ℛ is t-essentially coretractable ring, ℳ is faithful cyclic ℛ-module. Then ℳ is t- T-Essentially Coretractable Modules essentially coretractable The concept of t-essentially coretractable modules (8) A ring ℛ is t-essentially coretractable if and only are introduced with some of its properties. if for each proper I of ℛ, there exists r∈ ℛ, r≠0 such that r ∈annI and ≤tesℛ. Definition 1: A module ℳ is called t-essentially Proposition 1: Let ℳ be a nonsingular module. coretractable if for all proper submodule 퐾 of ℳ, Then ℳ is essentially coretractable if and only if it there exists 0≠f: ℳ/퐾→ ℳ such that is t-essentially coretractable. f(ℳ/퐾) ≤tes ℳ. A ring ℛ is called t-essentially Proof: (⇒) It is obvious since every essential coretractable if ℛ is a t-essentially coretractable ℛ- submodule is t-essential.. module. (⇐) Let 퐾 < ℳ, since ℳis t-essentially Examples and Remarks 1: coretractable, so ∃0≠f: ℳ/퐾 → ℳ and Imf≤tes ℳ. (1) It is clear that a module ℳ is t-essentially But ℳ is nonsingular hence Imf≤ess ℳ, therefore coretractable if and only if ∀ 퐾 < ℳ, there ℳ is essentially coretractable. exists f∈End(ℳ), f(퐾)=0 and Imf≤tes ℳ. Proposition 2: Let ℳ be a uniform module. If ℳ (2) For every ℛ-module, the following implications is coretractable, then ℳ is essentially coretractable are hold: and hence t-essentially coretractable. essentially coretractable module ⇒ t-essentially Proof: coretractable module ⇒ coretractable module. Let ℳ be a coretractable module and 퐾<ℳ, so The converse of each implication may be not ∃0≠f: ℳ/퐾→ ℳ that means Imf≠0. But ℳis hold, as the following examples show: uniform hence Imf≤essℳ, therefore ℳis essentially The Z-module Z6 is not essentially coretractable see coretractable. (6, Example(2.2(3))), but Z6 is coretractable "A module ℳis called d-Rickart if for each f ∈ ⊕ module. Also Z6 as Z-module t-essentially End(ℳ), Imf < ℳ" (9), so see the following: coretractable since for each N≤ Z6, ∃f∈End(Z6) Proposition 3: Every d-Rickart essentially and f(N)=0 and Imf≤tesZ6 (because coretractable module is epi-coretractable module. Imf+Z2(Z6)=Imf+Z6=Z6≤essZ6 and by (5, Proof: Let 퐾< ℳ, since ℳis essentially proposition(1.1)), Imf≤tesZ6). Beside these Z6 as coretractable, so ∃f: ℳ→ℳand Imf≤essℳ. But Z6-module is coretractable module, however it is ℳis d-Rickart, so Imf is a direct summand in not t-essentially coretractable since for each ℳ, therefore Imf=ℳ. Thus ℳ is epi-coretractable 0≠f∈End(Z6). Imf+Z2(Z6)= Imf+(0)=Imf≰ess Z6; module. that is Imf is not t-essential in Z6 by (5, Proposition 4: Let ℳ be Z2-torsion module (that is Proposition (1.1)). Z2(ℳ)= ℳ). Then ℳis coretractable if and only if (3) The two concepts t-essentially coretractable ℳis t-essentially coretractable. module and semisimple are independent, see the Proof: Since ℳis coretractable, so for all 퐾<ℳ, so following examples: The Z-module Z4 is t- ∃0≠f: ℳ/퐾→ ℳ. Then Imf≤ ℳ. But Imf+Z2(ℳ)= essentially coretractable since it is essentially Imf+ℳ=ℳ ≤essℳ so by (5,Proposition1.1) coretractable by (6, Example (2.2(4)), but Z4 is Imf≤tesℳ. Thus ℳis t-essentially coretractable not semisimple. module. The converse is clear. The Z6-module Z6 is semisimple but it is not t- By applying Proposition(4), see the following: essentially coretractable see Rem. & Exa. (2(3)). Let ℳ=Zn⨁Zm as Z-module for each n, m∈Z+. ℳis Also ℳ=Z2⨁Z2 as Z2-module is semisimple Z2-torsion module and coretractable, hence ℳis t- module but it is not t-essentially coretractable essentially coretractable. (4) Clearly every antihopfian module is t-essentially " A module ℳis t-semisimple if for every coretractable (since every antihopfian is submodule N of ℳthere exists a direct summand essentially coretractable (6) which implies t- 퐾 of ℳsuch that 퐾 is t-essential submodule of N" essentially coretractable). (5). (5) Every t-essentially coretractable module is C- Proposition 5: Let ℳbe a t-. coretractable, Y-coretractable module. The Then ℳis t-essentially coretractable if and only if converse is not true, see Z as Z-module is C-

157 Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986 for each 퐾<ℳ, ∃ f∈Hom(ℳ/퐾,ℳ), 퐼푚푓+푌 ℳ ℳ ℳ + 푍2( ) ≤ess . Thus Img≤tes by (5, Imf+Z (ℳ)= ℳ. 푌 푌 푌 푌 2 Proposition(1.1)). Proof: (⇒) If ℳ is t-essentially coretractable, so ' Corollary 1: Let f: ℳ → ℳ be an epimorphism for all 퐾<ℳ, so ∃0≠f: ℳ/퐾→ℳ, Imf≤ ℳ. Hence tes and kerf be y-closed submodule o푓ℳ. Then ℳ' is Imf+Z (ℳ) ≤ ℳ by (5, Proposition1.1) which 2 ess t-essentially coretractable module whenever ℳ is t- implies Imf+Z (ℳ)≤ ℳ. Thus Imf+Z (ℳ)= ℳ 2 tes 2 essentially coretractable. by ( (5, Corollary2.7)). ℳ Proof: By the 1st fundamental Theorem ≅ ℳ'. (⇐) By hypothesis for each K< ℳ, ∃f∈Hom(ℳ/퐾, 푘푒푟푓 ℳ ℳ), Im f + Z2(ℳ)= ℳ, hence Imf+Z2(ℳ)≤essℳ But is t-essentially coretractable module by 푘푒푟푓 and by (5, proposition 1.1), Imf≤tesℳ. It is to be noted that a direct summand of Proposition7. Thus ℳ' is t-essentially coretractable coretractable module need not be coretractable (12). module. Also it is to be noted that any direct summand of Corollary 2: Let ℳ be a t-essentially coretractable ℳ essentially coretractable module is essentially module. Then is t-essentially coretractable. 푍 (ℳ) coretractable see (6, corollary(2.7)), however if 2 Proof: Since 푍2(ℳ) is y-closed, then the required ℳ=C⊕N is a Z2-torsion R-module and C is a condition hold by Proposition(7). cogenerator (where an R-module ℳ is called a Corollary 3: Let ℳ=D⊕W be a t-essentially cogenerator if for any R-module N and 0≠x∈N, coretractable module such that 푍2(ℳ) ⊆W. Then there exists f: N→ ℳ such that f(x)≠0. (12)) and N D is t-essentially coretractable module. is any R-module, then ℳ is coretractable module Proof: Let ℳ=D⊕W . Since W is direct summand, by (12, proposition(1.5)) and so by proposition(6) so W is closed, but 푍2(ℳ) ⊆W by hypothesis and M is t-essentially coretractable but N need not be hence W is y-closed submodule. Hence by coretractable and hence N need not be t-essentially ℳ Proposition(7), is t-essentially coretractable coretractable. Recall that "A module ℳ is 푊 ℳ compressible if it can be embedded in any nonzero module but ≅ 퐷. Thus D is t-essentially 푊 its submodule"(13). coretractable module. Proposition 6: Let ℳ be a compressible t- Note that" a finite direct sum of coretractable essentially coretractable module and D be a direct modules is coretractable module", see(2, summand of ℳ, then D is t-essentially Proposition(2.6)). coretractable. Remark 2: The direct sum of t-essentially Proof: Since D≤⊕ ℳ, so ℳ=D⊕W for some coretractable modules need not be t-essentially W≤ ℳ. Let K< D, hence K⊕W≤ ℳ and so ∃0≠ coretractable module, for example: Let ℳ=Z2⊕Z2 ℳ ℳ 퐷 f: →ℳ with Imf≤ ℳ. Now since ≅ as Z2-module. ℳ is not t-essentially coretractable 퐾⊕W tes 퐾⊕W 퐾 and ℳ is compressible module, so ∃g: ℳ→D, g is but Z2 is t-essentially coretractable Z2-module. 퐷 퐷 Recall that" a module ℳ is called duo if every monomorphism. Then g◦f: →D and g◦f( )= g( 퐾 퐾 submodule of ℳ is fully invariant, where a 퐷 f( ) ). But Imf ≤ ℳ and g is monomorphism, so submodule N of ℳ is called fully invariant if for 퐾 tes 퐷 each f ∈ End(ℳ), f(N)⊆N"(15). "A submodule N g( f( )) ≤ D by (14, proposition(1.1.23)). Also 퐾 tes of an R-module ℳ is called stable if for each 퐷 퐷 g◦f≠0, because if g◦f=0 that is g◦f( )=0 so f( )=0 homomorphism f:N →ℳ, f(N) ≤ N"(16). 퐾 퐾 which is contradiction. Proposition 8: Let ℳ= ℳ1⊕ ℳ2 be a duo Proposition 7: Let Y be a y-closed submodule of t- module. Then ℳ is t-essentially coretractable if and ℳ essentially coretractable module ℳ. Then is t- only if ℳ1 and ℳ2 are t-essentially coretractable 푌 modules, provided that for each 퐾< ℳ, 퐾 ∩ essentially coretractable module. ℳ1<ℳ1 and 퐾 ∩ ℳ2 < ℳ2 . Proof: Let W/Y < ℳ/Y, hence W

158 Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986 g(y+퐾2)), h is a homomorphism. Imh= Corollary 4: Let ℳ be a finitely generated faithful ℳ ℳ1 ℳ2 multiplication module over a commutative ring ℛ. Imf⊕Img≤tes ℳ1 ⊕ ℳ2= ℳ, but ≅( ⊕ ) 퐾 퐾1 퐾2 Then the following are equivalent: see Kasch hence there exists an isomorphism α, (1) ℳ is t-essentially coretractable module; ℳ ℳ ℳ ℳ where α: ⟶ 1 ⊕ 2 and so α◦h: ⟶ ℳ. As (2) ℛ is t-essentially coretractable ring; 퐾 퐾 퐾 퐾 1 2 (3) 퐸푛푑(ℳ) is t-essentially coretractable ring. Im(α◦h)= α(Imh), Imh≤ ℳ and α is isomorphism tes Proof: (1)⟺(2) It holds by proposition(9). so that α(Imh) ≤ ℳ that is tes (2)⟺(3) Since ℳ is finitely generated Im(α◦h) ≤ ℳ= ℳ ℳ . Thus ℳ is t-essentially tes 1 2 faithful multiplication module, ∀ f∈End(ℳ) coretractable. ∃0 ≠r ∈ ℛ and f(m)=mr ∀m ∈ ℳ. Define 휑: ℛ → (⇒) To prove ℳ is t-essentially coretractable 1 End(ℳ) by 휑(r)=f if f(m)=mr ∀m∈ ℳ. 휑 is well- module. Let 퐾< ℳ .Then K⊕ ℳ < ℳ and since 1 2 defined and epimorphism, but kerf=annℳ=0. Thus ℳ is t-essentially coretractable, ∃0≠f∈End(ℳ) and End(ℳ)≅ ℛ/annℳ ≅ ℛ, therefore End(ℳ) is t- f(K⊕ ℳ )=0 and Imf≤ ℳ. Now f(K⊕ ℳ )=0 2 tes 2 essentially coretractable. implies to f(K)=0 and f(ℳ )=0. Let 2 g=f| : ℳ →ℳ. Since ℳ is a fully invariant ℳ1 1 1 Weakly t-Essentially Coretractable Modules direct summand of ℳ, ℳ1 is stable; that is Definition 2: A module ℳ is called weakly t- g(ℳ1) ⊆ ℳ1 and hence f(ℳ1)⊆ ℳ1. It follows that essentially coretractable if ∀ W< ℳ, g(ℳ1)=f(ℳ1)=f(ℳ1⊕ ℳ2)=f(ℳ)≤tesℳ= ℳ1⊕ ∃ 0 ≠f: ℳ/W→ℳ such that f(ℳ /W)+W ≤tes ℳ. ℳ2. Thus g(ℳ1) ⊕(0) ≤tesℳ1⊕ ℳ2 which implies A ring ℛ is called weakly t-essentially coretractable to g(ℳ1)≤tesℳ1. Also g(K)=f(K)=0. Thus ℳ1 is t- if ℛ is a weakly t-essentially coretractable ℛ- essentially coretractable module. module. Recall that" an ℛ-module ℳ is called a Examples and Remarks 3: multiplication module if for each submodule N of (1) A module ℳ is weakly t-essentially ℳ, there exists a right ideal in R such that coretractable if and only if ∀ W< ℳ, ℳI=N"(17). ∃ f∈End(ℳ), f(W)=0 and Imf+W≤tesℳ. Proposition 9: Let ℳbe a finitely generated Proof: Clear. faithful multiplication module. Then ℳ is t- (2) It is clear that any essentially coretractable essentially coretractable if and only if ℛ is t- module is weakly t-essentially coretractable. essentially coretractable ring, where ℛ is a The converse of this part is not true, consider commutative ring. the following: Z12 as Z-module is weakly t- Proof: (⇒) Let I < ℛ. Then N= ℳI < ℳ since ℳ is essentially coretractable and not essentially a finitely generated faithful multiplication module. coretractable see (6). As ℳ is t-essentially coretractable, ∃ f∈End(ℳ), (3) Every weakly t-essentially coretractable f(N)=0 and Imf ≤tes ℳ. But ℳis finitely generated module is coretractable module. multiplication module, so ∃0 ≠r ∈ ℛ such that (4) By Rem. & Exa. (1(3)), the semisimple and t- f(m)=mr ∀m∈ ℳ. Define g: ℛ → ℛ by g(a)=ar essentially coretractable modules are ∀a∈ ℛ. Clearly g is an ℛ-homomorphism and independent. However a semisimple module is g(I)=Ir and g(ℛ)=ℛ푟. Since weakly t-essentially coretractable. f(N)=f(ℳI)=f(ℳ)I=ℳrI=ℳIr=0, then Ir⊆annℳ=0 Proof: Let W<ℳ. ℳis semisimple module, so and so Ir=0, that is g(I)=0. Also Imf=ℳ≤tesℳ ∃ D≤⨁ ℳ, W⨁D=ℳ hence ℳ/W≅D. Let implies that ≤tes ℛ by (14, Lemma(1.1.24)). f=i◦g, where i is the inclusion map from D into Thus g(ℛ) ≤tesℛ. Hence ℛ is t-essentially M and g is an isomorphism between ℳ/W and coretractable. D. It is clear that f≠0 and Im f +W≤tesℳ. (⇐) Let N<ℳ. Since ℳ is a multiplication module, Therefore ℳis weakly t-essentially then N=ℳI for some I≤ℛ. But ℳ is a finitely coretractable module. generated faithful multiplication module, so I<ℛ. By applying (4), consider Z2⨁Z2 as Z2- As ℛ is t-essentially coretractable, so by Rem. & module is weakly t-essentially coretractable, Exa. (1(9)), ∃r∈ ℛ, r≠0 such that r∈annI and but not t-essentially coretractable module. Also ≤tesℛ. Now, define f: ℳ → ℳ by f(m)=mr the converse may not be true, for example see ∀m∈ ℳ. It is clear that f is an ℛ-homomorphism, Z12 as Z-module is weakly t-essentially f(N)=f(ℳ퐼)=f(ℳ)I=ℳrI=ℳIr=0 and f(ℳ)= ℳ푟= coretractable but not semisimple module. ℳ since ≤tes ℛ, then by (14, (5) It is clear that any module over semisimple Lemma(1.1.24)), ℳ푟 ≤tesℳ; that is f(ℳ) ≤tesℳ. ring ℛ is semisimple and hence it is weakly t- Thus ℳ is t-essentially coretractable. essentially coretractable module by Rem. & Exa. (3(4)).

159 Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986

(6) A module ℳ is weakly t-essentially direct summand of ℳ, then D is weakly t- coretractable ℛ-module if and only if ℳ is essentially coretractable module. weakly t-essentially coretractable ℛ̅-module Proof: Since D ≤⊕ ℳ, so ℳ=D ⊕W for some (where, ℛ̅=ℛ/annℳ). W≤ ℳ. Let K+I≤tesℛ. Proof: Since ℳ is duo, ℳ1 and ℳ2 are fully Proposition 10: Let ℳ be a nonsingular uniform invariant submodules, but ℳ1 and ℳ2 are direct module. Then the following statements are summand, then by Proposition(12), ℳ1 and ℳ2 are equivalent: weakly t-essentially coretractable modules.

(1) ℳis essentially coretractable; Corollary 6: Let ℳ=ℳ1 ⊕ ℳ2 with (2) is t-essentially coretractable; ℳ annℳ1+annℳ2= ℛ. If ℳ is weakly t-essentially (3) is weakly t-essentially coretractable; ℳ coretractable module, then ℳ1 and ℳ2 are weakly (4) ℳis weakly essentially coretractable. t-essentially coretractable. Proof: (1)⟺(2) It follows by Proposition Proof: Since annℳ1+annℳ2=R, then (1). ℳ1=ℳ1annℳ2 and ℳ2= ℳ2annℳ1. Hence for (2) ⇒(3) Let W<ℳ. Since ℳis t- each f:ℳ ⟶ ℳ, f(ℳ1)=f(ℳ1)annℳ2⊆ essentially coretractable, so ∃ f∈End(ℳ), ℳannℳ2=ℳ1annℳ2=ℳ1 hence ℳ1 is a fully f(W)=0, Imf≤tesℳ and hence invariant. Similarity ℳ2 is a fully invariant. But ℳ1 Imf+W≤tesℳ. Thus ℳis weakly t- and ℳ2 are direct summands. Thus they are weakly essentially coretractable. t-essentially coretractable modules by Proposition (3)⟺(4) Let 퐾 < ℳ, since ℳis weakly t- (12). essentially coretractable, so ∃f: ℳ/퐾→ ℳ Recall that " an ℛ-module ℳ is called a and Imf+K ≤tes ℳ. But ℳ is nonsingular polyform if for any submodule L ⊆ ℳ and for any hence Imf+K≤ess ℳ, therefore ℳ is 0≠ 휑:L → ℳ, Ker 휑 is not essential in L. weakly essentially coretractable Equivalently, if for any submodule L ⊆ ℳ and for (4)⟺(1) Since ℳ is nonsingular any 휑: L→ ℳ, Ker휑 ≤e L implies 휑= 0 "(9). uniform, then ℳ is quasi-Dedekind by (18, Proposition 13: Let ℳbe a polyform module. If ℳ̅ Proposition(1.5)), and hence (4)⟺(1) by ( where ℳ̅ is the quasi- of ℳ ) is using (6, Proposition(3.9)). weakly t-essentially coretractable module, then ℳis Proposition 11: Let ℳ be Z2-torsion R-module. weakly t-essentially coretractable Then the following are equivalent: Proof: Since ℳ is polyform, so End(ℳ̅ ) is regular (1) ℳis coretractable; ring by (9, Theorem(2.1)). But ℳ̅ is coretractable (2) ℳis t-essentially coretractable; hence by (3, Proposition(2.1)), ℳ̅ is semisimple (3) ℳis weakly t-essentially coretractable and hence ℳ is semisimple. Thus ℳ is weakly t- Proof: (1) ⟺ (2) It follows by proposition(4). essentially coretractable module. (2) ⇒ (3) It is clear. It is known every nonsingular module is (3) ⇒(1) It follows by Rem. & Exa. (3(3)). polyform, so one can get the following directly: Proposition 12: Let ℳ be a weakly t-essentially Corollary 7: Let ℳ be a nonsingular module. If coretractable module and D be a fully invariant ℳ̅ is weakly t-essentially coretractable module, then ℳis weakly t-essentially coretractable.

160 Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986

Proposition 14: Let ℳ be a nonsingular ℛ-module. and hence by Rem. & Exa.(4(2)), ℛ is Then the following statements are equivalent: completely weakly t-essentially coretractable (1) ℳ is coretractable; ring. (2) ℳ is semisimple; (5) Since a dual Rickart ring is semisimple (2), (3) ℳ is weakly t-essentially coretractable; then every dual Rickart ring is completely (4) ℳ is weakly essentially coretractable. weakly t-essentially coretractable ring by Rem. Proof: (1) ⇒ (2) By (12, proposition 1.3). & Exa. (4(2)). (2) ⇒ (3) By Rem. & Exa. (3(3)). Proposition 17: Let ℛ be a commutative ring such (3) ⇒(4) It follows by Rem. & Exa. (3(7)). that ℛ̅= ℛ/annℳ is a weakly t-essentially (4) ⇒(1) It is clear. coretractable ℛ-module. Then every cyclic ℛ- Proposition 15: Let ℳbe a finitely generated module is weakly t-essentially coretractable faithful multiplication module. Then ℳ is weakly t- Proof: Let ℳbe a cyclic ℛ-module. Then ℳ ≅ ℛ̅ essentially coretractable if and only if ℛ is weakly is weakly t-essentially coretractable ℛ-module. t-essentially coretractable ring, where ℛ is a Corollary 9: Let ℛ be a commutative ring. Then commutative ring. the following are equivalent: Proof: It is similar to the proof of proposition(9). (1) ℛ is completely weakly t-essentially Corollary 8: Let ℳbe a finitely generated faithful coretractable ; multiplication module over a commutative ring ℛ. (2) Every cyclic ℛ-module is weakly t- Then the following are equivalent: essentially coretractable (1) ℳ is weakly t-essentially coretractable (3) ℛ/ann ℳ is weakly t-essentially module; coretractable ℛ-module. (2) ℛ is weakly t-essentially coretractable ring; Proposition 18: Let ℛ be a ring with J(ℛ)=0. Then (3) 퐸푛푑(ℳ) is weakly t-essentially the following are equivalent: coretractable ring. (1) ℛ is weakly t-essentially coretractable ring; Proposition 16: Let ℳ be a weakly t-essentially (2) ℛ is completely weakly t-essentially coretractable and t-semisimple ℛ-module such that coretractable ring; N⊆ Z2(ℳ), ∀N< ℳ. Then ℳis t-essentially (3) ℛ is weakly t-essentially coretractable ring; coretractable. (4) All free ℛ-module is weakly t-essentially Proof: Let N< ℳ. Since ℳ is weakly t-essentially coretractable; coretractable ℛ-module, ∃ f∈End(ℳ), f(N)=0 and (5) All finitely generated free ℛ-module are Imf+N≤tes ℳ. Then by (5, proposition(1.1)), weakly t-essentially coretractable. Imf+N+Z2 (ℳ)≤ess ℳ. Since N⊆Z2(ℳ), then Proof: (1) ⇒ (3) Since ℛ is weakly t-essentially Imf+Z2(ℳ)≤ess ℳ. Again by (5,proposition(1.1)), coretractable ring, ℛ is coretractable and hence by Imf≤tes ℳ. Thus ℳ is t-essentially coretractable (2, Lemma(3.7)), ℛ is semisimple. module (3) ⇒ (2) and (3) ⇒ (4) are clear by Rem. & Recall that "a ring ℛ is called completely Exa.(4(2)). coretractable ring (briefly, CC-ring) if every ℛ- (4) ⇒ (5) It is clear. module is coretractable"(2). See the following: (5) ⇒ (1) It is clear, since ℛ is a finitely generated Definition 3: A ring ℛ is called completely weakly free ℛ-module. t-essentially coretractable ring if every ℛ-module is Remark 5: Since every commutative regular ring ℛ weakly t-essentially coretractable. (in the sense of Von Neumann) satisfies J(ℛ)=0, Remarks and Examples 4: then proposition(18) is hold for commutative (1) It clear that, every completely weakly t- regular rings. essentially coretractable ring is CC-ring. Corollary 11: If ℛ is a commutative regular (in the (2) By Rem. & Exa. (3(5)), every semisimple ring sense of Von Neumann) weakly t-essentially is completely weakly t-essentially coretractable coretractable ring, then ℛ is a principal ideal ring ring. (PIR). (3) Recall that" a ring ℛ is called Kasch if every Proof: By previous proposition, ℛ is semisimple. simple ℛ-module can be embedded in ℛ"(2). Hence every ideal generated by idempotent If ℛ is Kasch ring and J(ℛ)=0, then ℛ is element. Thus ℛ is a principal ideal ring. semisimple (2) and hence by Rem. & Exa.(4(2)), ℛ is completely weakly t- Conclusion: essentially coretractable ring. In this paper, the notions of t-essentially and (4) If ℛ is Kasch ring and regular (in the sense of weakly t-essentially coretractable modules are Von Neumann), then ℛ is semisimple by (2) defined as a generalization of essentially and

161 Open Access Baghdad Science Journal P-ISSN: 2078-8665 Published Online First: December 2020 E-ISSN: 2411-7986 weakly essentially coretractable module. Also, 7. Hadi IMA, Al-Aeashi SN. Y-coretractable and several results are given. Further the completely strongly Y-coretractable modules. Asian J. Applied weakly t-essentially coretractable rings are defined Sci . 2017; 5(2): 427-433. and investigated. 8. Gang Y, Zhong-kui L. On hopfian and co-hopfian modules. Vietnam J. Math . 2007; 35: 73–80. 9. Hadi IMA, Al-Aeashi SN. The class of coretractable modules and some generalizations. GJPAM. 2018; Authors' declaration: 12(2): 811-820. - Conflicts of Interest: None. 10. Naoum AG, Al-Hashimi BA, Yaseen S M. Coquasi- - We hereby confirm that all the Figures and invertible submodule. Iraqi J. of science. 2006; 47(1): Tables in the manuscript are mine ours. Besides, 154-159. DOI: 10.24996.ijs.58.2C.10. the Figures and images, which are not mine ours, 11. Hadi IMA, Al-Aeashi SN. C-coretractable and have been given the permission for re- strongly C-coretractable modules. Special Issus: 1st publication attached with the manuscript. Scientific International Conference, College of Science, Al-Nahrain University. 2019: 65-72. - Ethical Clearance: The project was approved by 12. Hadi IMA, Al-Aeashi SN. Weakly coretrectable the local ethical committee in University of modules. J. Phys.: Conf. Ser. 2018; 1003 012061. Doi Baghdad. :10.1088/1742-6596/1003/1/012061. 13. Nguyen DHN, Ronnason C. On slightly References: compressible-injective modules. EJPAM 2018; DOI: 1. Clark J, Lomp C, Vanaja N, Wisbauer R. Lifting 10.29020/nybg.ejpam.v11i3.3291 . modules. Supplements and Projectivity in Module 14. Shyaa FD, A study of modules with related T- Theory. Springer Science & Business Media; 2008 semisimple modules, Ph.D. Thesis, University of Aug 17. Baghdad, Iraq, 2018. 2. Amini B, Ershad M, Sharif H. Coretractable modules. 15. Ozcan A, Harmanci H, Smith PF. Duo modules. J. Aust. Math. Soc. 2009; 86(3): 289–304. Glasgow Math. J. 2006; 48(3): 533-545. 3. Hadi IMA, Al-Aeashi SN. Strongly coretractable 16. Abbas MS, Salman HA. Fully small dual stable modules. Iraqi J. of science. 2017; 58(2C):1069-1075. modules. IJS. 2018; 59(1B): 398-403. DOI: 4. Goodearl KR. , nonsingular rings and 10.24996/ijs.2018.59.1B.19. modules. New York, Marcel Dekker, 1976. 17. El-Bast ZA, Smith PF.Multiplication modules. 5. Asgari Sh, Haghany A, Tolooei Y. T-semisimple Comm. Algebra. 1988;16:755–779. modules and T-semisimple rings. Comm. in Algebra. 18. Hadi IMA, Al-Aeashi SN. Some results about 2013; 41(5): 1882-1902. coretractable modules. Journal of AL-Qadisiyah for 6. Al-Aeashi SN, Shyaa FD, Hadi IMA. Essentially and computer science and mathematics. 2017; 9(2): 40- weakly essentially coretractable modules. J. Phys.: 48. Conf. Ser. 2019; 1234, 012107. Doi:10.1088/1742- 19. Al-Saadi SA. S-extending modules and related 6596/1234/1/012107. concepts. Ph.D. Thesis, University of Al- Mustansiriya, Baghdad, Iraq, 2007.

المقاسات المنكمشة المضادة الجوهرية والجوهرية الضعيفة من النوع-T

انعام محمد علي هادي1 شكر نعمة العياشي2 فرحان داخل شياع3

1 قسم الرياضيات, كلية التربية للعلوم الصرفة)ابن الهيثم(, جامعة بغداد. 2 قسم الرياضيات, كلية التربية, جامعة الكوفة, الكوفة, العراق. 3 قسم الرياضيات, كلية التربية, جامعة القادسية, القادسية, العراق.

الخالصة: نقدم في هذا البحث اعمامات جديدة للمقاسات المنكمشة المضادة حيث اطلقنا على المقاس ℳ اسم منكمش مضاد جوهري من النوع-T أو منكمش مضاد جوهري ضعيف من النوع-T إذا كان لكل مقاس جزئي K من المقاس ℳ , يوجد (f∈End(ℳ و f (K)=0 بحيث ان Imf≤tes ℳ او Imf + K≤tes ℳ. تم دراسة بعض الخصائص األساسية و عرض العديد من العالقات بين هذه المقاسات ومقاسات اخرى ذات الصلة.

الكلمات المفتاحية: وحدة قابلة للتجديد , وحدة قابلة للتجزئة بشكل أساسي, وحدة قابلة للتجديد بشكل أساسي, وحدة قابلة للتجديد بشكل ضعيف, وحدة قابلة للتجميع بشكل أساسي.

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