Injective property relative to nonsingular exact sequences

Bull. Korean Math. Soc. 2017 Vol. 54, No. 2, 559-571 https://doi.org/10.4134/BKMS.b160184 Published online November 11, 2016 Printed March 31, 2017

Marzieh Arabi-Kakavand, Shadi Asgari, and Yaser Tolooei Isfahan University of Technology, Institute for Research in Fundamental Sciences (IPM), Razi University

Abstract : We investigate modules $M$ having the injective property relative to nonsingular modules. Such modules are called ``$\mathcal N$-injective modules''. It is shown that $M$ is an $\mathcal N$-injective $R$-module if and only if the annihilator of $Z_2(R_R)$ in $M$ is equal to the annihilator of $Z_2(R_R)$ in $E(M)$. Every $\mathcal N$-injective $R$-module is injective precisely when $R$ is a right nonsingular ring. We prove that the endomorphism ring of an $\mathcal N$-injective module has a von Neumann regular factor ring. Every (finitely generated, cyclic, free) $R$-module is $\mathcal N$-injective, if and only if $R^{(\mathbb N)}$ is $\mathcal N$-injective, if and only if $R$ is right $t$-semisimple. The $\mathcal N$-injective property is characterized for right extending rings, semilocal rings and rings of finite reduced rank. Using the $\mathcal N$-injective property, we determine the rings whose all nonsingular cyclic modules are injective.

Keywords : nonsingular and $Z_2$-torsion modules, $\mathcal N$-injective modules, right $t$-semisimple rings