Mathematics > Home > Advances in Pure and Applied Mathematics > Issue 1 > Article

Peter Danchev

Bulgarian Academy of Sciences

Bulgaria

Mahdi Samiei

Velayat University

Iran

Published on 3 July 2020 DOI : 10.21494/ISTE.OP.2020.0542

A commutative ring *R* is called *J-rad clean* in case, for any *r ∈ R*, there is an idempotent *e ∈ R* such that *r−e ∈ U(R)* and *re ∈ J(R)*, where *U(R)* and *J(R)* denote the set of units and the Jacobson radical of *R*, respectively. Also, a ring R is called semiregular if *R/J(R)* is regular in the sense of von Neumann and idempotents lift modulo *J(R)*. We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring *RG* is *J*-rad clean if, and only if, *R* is a commutative *J*-rad clean ring and *G* is a torsion abelian group, provided that *J(R)* is nil.

A commutative ring *R* is called *J-rad clean* in case, for any *r ∈ R*, there is an idempotent *e ∈ R* such that *r−e ∈ U(R)* and *re ∈ J(R)*, where *U(R)* and *J(R)* denote the set of units and the Jacobson radical of *R*, respectively. Also, a ring R is called semiregular if *R/J(R)* is regular in the sense of von Neumann and idempotents lift modulo *J(R)*. We demonstrate that these two concepts are, actually, equivalent and we portray a portion of the properties of this class of rings. In particular, as a direct application, we prove that the commutative group ring *RG* is *J*-rad clean if, and only if, *R* is a commutative *J*-rad clean ring and *G* is a torsion abelian group, provided that *J(R)* is nil.