Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Subject: search.filters.subject.General module theory

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  • Article
    Citation - WoS: 3
    Citation - Scopus: 3
    Small Supplements, Weak Supplements and Proper Classes
    (Hacettepe Üniversitesi, 2016) Alizade, Rafail; Büyükaşık, Engin; Durğun, Yılmaz
    Let SS denote the class of short exact sequences E:0 → Af→ B → C → 0 of R-modules and R-module homomorphisms such that f(A) has a small supplement in B i.e. there exists a submodule K of M such that f(A) + K = B and f(A) ∩ K is a small module. It is shown that, SS is a proper class over left hereditary rings. Moreover, in this case, the proper class SS coincides with the smallest proper class containing the class of short exact sequences determined by weak supplement submodules. The homological objects, such as, SS-projective and SScoinjective modules are investigated. In order to describe the class SS, we investigate small supplemented modules, i.e. the modules each of whose submodule has a small supplement. Besides proving some closure properties of small supplemented modules, we also give a complete characterization of these modules over Dedekind domains.
  • Article
    Absolute Co-Supplement and Absolute Co-Coclosed Modules
    (Hacettepe Üniversitesi, 2013) Tütüncü, Derya Keskin; Toksoy, Sultan Eylem
    A module M is called an absolute co-coclosed (absolute co-supplement) module if whenever M ≅ T/X the submodule X of T is a coclosed (supplement) submodule of T. Rings for which all modules are absolute co-coclosed (absolute co-supplement) are precisely determined. We also investigate the rings whose (finitely generated) absolute co-supplement modules are projective. We show that a commutative domain R is a Dedekind domain if and only if every submodule of an absolute co-supplement R-module is absolute co-supplement. We also prove that the class Coclosed of all short exact sequences 0→A→B→C→0 such that A is a coclosed submodule of B is a proper class and every extension of an absolute co-coclosed module by an absolute co-coclosed module is absolute co-coclosed.
  • Article
    Citation - WoS: 28
    Citation - Scopus: 29
    Rad-Supplemented Modules
    (Universita di Padova, 2010) Büyükaşık, Engin; Mermut, Engin; Özdemir, Salahattin
    Let τ be a radical for the category of left R-modules for a ring R. If M is a τ-coatomic module, that is, if M has no nonzero τ-torsion factor module, then τ(M) is small in M. If V is a τ-supplement in M, then the intersection of V and τ(M) is τ(V). In particular, if V is a Rad-supplement in M, then the intersection of V and Rad(M) is Rad(V). A module M is τ-supplemented if and only if the factor module of M by P τ(M) is τ-supplemented where P τ(M) is the sum of all τ-torsion submodules of M. Every left R-module is Rad-supplemented if and only if the direct sum of countably many copies of R is a Rad-supplemented left R-module if and only if every reduced left R-module is supplemented if and only if R/P(R) is left perfect where P(R) is the sum of all left ideals I of R such that Rad I = I. For a left duo ring R, R is a Rad-supplemented left R-module if and only if R/P(R) is semiperfect. For a Dedekind domain R, an R-module M is Rad-supplemented if and only if M/D is supplemented where D is the divisible part of M.