27 research outputs found

    Module decompositions via Rickart modules

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    This work is devoted to the investigation of module decompositions which arise from Rickart modules, socle and radical of modules. In this regard, the structure and several illustrative examples of inverse split modules relative to the socle and radical are given. It is shown that a module M has decompositions M = Soc(M) ⊕ N and M = Rad(M) ⊕ K where N and K are Rickart if and only if M is Soc(M)-inverse split and Rad(M)-inverse split, respectively. Right Soc(·)-inverse split left perfect rings and semiprimitive right hereditary rings are determined exactly. Also, some characterizations for a ring R which has a decomposition R = Soc(RR) ⊕ I with I a hereditary Rickart module are obtained

    Symmetric modules over their endomorphism rings

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    Let R be an arbitrary ring with identity and M a right R-module with S=EndR(M). In this paper, we study right R-modules M having the property for f,g∈EndR(M) and for m∈M, the condition fgm=0 implies gfm=0. We prove that some results of symmetric rings can be extended to symmetric modules for this general setting

    Symmetric modules over their endomorphism rings

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    Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). In this paper, we study right R-modules M having the property for f, g ∈ EndR(M) and for m ∈ M, the condition fgm = 0 implies gfm = 0. We prove that some results of symmetric rings can be extended to symmetric modules for this general setting. © Journal “Algebra and Discrete Mathematics”

    Generalized symmetric rings

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    In this paper, we introduce a class of rings which is a generalization of symmetric rings. Let R be a ring with identity. A ring R is called central symmetric if for any a, b,c∈R, abc=0 implies bac belongs to the center of R. Since every symmetric ring is central symmetric, we study sufficient conditions for central symmetric rings to be symmetric. We prove that some results of symmetric rings can be extended to central symmetric rings for this general settings. We show that every central reduced ring is central symmetric, every central symmetric ring is central reversible, central semmicommutative, 2-primal, abelian and so directly finite. It is proven that the polynomial ring R[x] is central symmetric if and only if the Laurent polynomial ring R[x,x−1] is central symmetric. Among others, it is shown that for a right principally projective ring R, R is central symmetric if and only if R[x]/(xn) is central Armendariz, where n≥2 is a natural number and (xn) is the ideal generated by x

    A generalization of reduced rings

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    Let R be a ring with identity. We introduce a class of rings which is a generalization of reduced rings. A ring R is called central rigid if for any a, b ? R, a2b = 0 implies ab belongs to the center of R. Since every reduced ring is central rigid, we study sufficient conditions for central rigid rings to be reduced. We prove that some results of reduced rings can be extended to central rigid rings for this general setting, in particular, it is shown that every reduced ring is central rigid, every central rigid ring is central reversible, central semicommutative, 2-primal, abelian and so directly finite

    Central quasipolar rings

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    In this paper, we introduce a kind of quasipolarity notion for rings, namely, an element a of a ring R is called central quasipolar if there exists p2 = p ∈ R such that a + p is central in R, and the ring R is called central quasipolar if every element of R is central quasipolar. We give many characterizations and investigate general properties of central quasipolar rings. We determine the conditions that some subrings of upper triangular matrix rings are central quasipolar. A diagonal matrix over a local ring is characterized in terms of being central quasipolar. We prove that the class of central quasipolar rings lies between the classes of commutative rings and Dedekind nite rings, and a ring R is central quasipolar if and only if it is central clean. Further we show that several results of quasipolar rings can be extended to central quasipolar rings in this general setting

    A generalization of reversible rings

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    WOS: 000344830500006In this paper, we introduce a class of rings which is a generalization of reversible rings. Let R be a ring with identity. A ring R is called central reversible if for any a,b is an element of R, ab=0 implies ba belongs to the center of R. Since every reversible ring is central reversible, we study sufficient conditions for central reversible rings to be reversible. We prove that some results of reversible rings can be extended to central reversible rings for these general settings

    Strongly Large Module Extensions

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    In this paper, we introduce strongly large submodules and investigate their properties. A submodule N of a right R-module M is said to be strongly large in case for any m is an element of M, s is an element of R with ms not equal 0 there exists an r is an element of R such that mr is an element of N and mrs not equal 0. In this note, we also define and study strongly large closed submodules and strongly large complement submodules.WoSScopu
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