Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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INJECTIVE PROPERTY RELATIVE TO NONSINGULAR EXACT SEQUENCES

  • Arabi-Kakavand, Marzieh (Department of Mathematical Sciences Isfahan University of Technology) ;
  • Asgari, Shadi (Department of Mathematical Sciences University of Isfahan) ;
  • Tolooei, Yaser (Department of Mathematics Faculty of Science Razi University)
  • Received : 2016.03.03
  • Published : 2017.03.31
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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.

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