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 module 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, singular submodule 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 ideal I of ℛ, there exists r∈ ℛ, r≠0 such that r ∈annI and <r>≤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-semisimple module. 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 .