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    Generalizations Of Second Submodules
    (Univ Nis, Fac Sci Math, 2020) Gezen, Secil Ceken; Tekir, Unsal; Koc, Suat; Kelleqi, Ajlin
    In this paper, we introduce and study some new generalizations of second submodules via a function phi on the set of all submodules of a module. Let R be a ring with non-zero identity, M be an R-module and phi : S(M) -> S(M) be a function where S(M) is the set of all submodules of M. A non-zero submodule N of M is said to be a phi-second submodule if, for any element a of R and a submodule K of M, aN subset of K and a phi(N) not subset of K imply either N subset of K or a is an element of ann(R)(N). Let n >= 2 be an integer and phi(n) : S(M) -> S(M) be the function defined by phi(n)(L) = (L : (M) ann(R)(L)(n-1)) for every L is an element of S(M). Then a phi(n)-second submodule of M is said to be an n-almost second submodule of M. We determine various algebraic properties of these submodules and investigate their relationships with other known submodule classes such as second, prime and semisimple submodules. We study the structure of n-almost second submodules of modules over ZPI-rings and Dedekind domains. We also give some characterizations of modules and submodules by using n-almost second submodules.
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    Öğe
    On normal modules
    (Taylor & Francis Inc, 2023) Jayaram, Chillumuntala; Tekir, Unsal; Koc, Suat; Ceken, Secil
    Recall that a commutative ring R is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module M is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules.