• Nazim Agayev

      Articles written in Proceedings – Mathematical Sciences

    • On a Class of Semicommutative Modules

      Nazim Agayev Tahire Özen Abdullah Harmanci

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      Let 𝑅 be a ring with identity, 𝑀 a right 𝑅-module and $S=\mathrm{End}_R(M)$. In this note, we introduce 𝑆-semicommutative, 𝑆-Baer, $S-q.$-Baer and $S-p.q.$-Baer modules. We study the relations between these classes of modules. Also we prove if 𝑀 is an 𝑆-semicommutative module, then 𝑀 is an $S-p.q.$-Baer module if and only if $M[x]$ is an $S[x]-p.q.$-Baer module, 𝑀 is an 𝑆-Baer module if and only if $M[x]$ is an $S[x]$-Baer module, 𝑀 is an $S-q.$-Baer module if and only if $M[x]$ is an $S[x]-q.$-Baer module.

    • On a generalization of semisimple modules

      NAZIM AGAYEV CESIM ÇELIK TAHIRE ÖZEN

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      Let $R$ be a ring with identity. A module $M_{R}$ is called an $r$-semisimple module if for any right ideal $I$ of $R$, $M I$ is a direct summand of $M_{R}$ which is a generalization of semisimple and second modules. We investigate when an $r$-semisimple ring is semisimple and prove that a ring $R$ with the number of nonzero proper ideals $\leq 4$ and $J (R) = 0$ is $r$-semisimple. Moreover, we prove that $R$ is an $r$-semisimple ring if and only if it is a direct sum of simple rings and we investigate the structure of module whenever $R$ is an $r$-semisimple ring.

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