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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2018, Volume 26, Issue 2, Pages 170–189 (Mi adm679)

RESEARCH ARTICLE

Modules in which every surjective endomorphism has a $\delta$-small kernel

Shahabaddin Ebrahimi Atani, Mehdi Khoramdel, Saboura Dolati Pishhesari

Department of Mathematics, University of Guilan, P.O.Box 1914, Rasht, Iran

Abstract: In this paper, we introduce the notion of $\delta$-Hopfian modules. We give some properties of these modules and provide a characterization of semisimple rings in terms of $\delta$-Hopfian modules by proving that a ring $R$ is semisimple if and only if every $R$-module is $\delta$-Hopfian. Also, we show that for a ring $R$, $\delta(R)=J(R)$ if and only if for all $R$-modules, the conditions $\delta$-Hopfian and generalized Hopfian are equivalent. Moreover, we prove that $\delta$-Hopfian property is a Morita invariant. Further, the $\delta$-Hopficity of modules over truncated polynomial and triangular matrix rings are considered.

Keywords: Dedekind finite modules, Hopfian modules, generalized Hopfian modules, $\delta$-Hopfian modules.

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MSC: 16D10, 16D40, 16D90
Revised: 18.10.2018
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Citation: Shahabaddin Ebrahimi Atani, Mehdi Khoramdel, Saboura Dolati Pishhesari, “Modules in which every surjective endomorphism has a $\delta$-small kernel”, Algebra Discrete Math., 26:2 (2018), 170–189

Citation in format AMSBIB
\Bibitem{AtaKhoPis18} \by Shahabaddin~Ebrahimi~Atani, Mehdi~Khoramdel, Saboura~Dolati~Pishhesari \paper Modules in which every surjective endomorphism has a $\delta$-small kernel \jour Algebra Discrete Math. \yr 2018 \vol 26 \issue 2 \pages 170--189 \mathnet{http://mi.mathnet.ru/adm679}