Open Access
2013 On rings whose modules have nonzero homomorphisms to nonzero submodules
Y. Tolooei, M. R. Vedadi
Publ. Mat. 57(1): 107-122 (2013).


We carry out a study of rings $R$ for which $\operatorname{Hom}_R(M,N)\neq 0$ for all nonzero $ N\leq M_R$. Such rings are called retractable. For a retractable ring, Artinian condition and having Krull dimension are equivalent. Furthermore, a right Artinian ring in which prime ideals commute is precisely a right Noetherian retractable ring. Retractable rings are characterized in several ways. They form a class of rings that properly lies between the class of pseudo-Frobenius rings, and the class of max divisible rings for which the converse of Schur's lemma holds. For several types of rings, including commutative rings, retractability is equivalent to semi-Artinian condition. We show that a Köthe ring $R$ is an Artinian principal ideal ring if and only if it is a certain retractable ring, and determine when $R$ is retractable.


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Y. Tolooei. M. R. Vedadi. "On rings whose modules have nonzero homomorphisms to nonzero submodules." Publ. Mat. 57 (1) 107 - 122, 2013.


Published: 2013
First available in Project Euclid: 18 December 2012

zbMATH: 1300.16009
MathSciNet: MR3058929

Primary: 16D10
Secondary: 16A55 , 16E50 , 16L30

Keywords: CPF rings , max ring , regular ring , retractable , semi-Artinian

Rights: Copyright © 2013 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.57 • No. 1 • 2013
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