Abstract
In this paper several new properties of singular ideals and non- singular rings are obtained, a connection between a singular ideal and the Jacobson radical is considered , and a sufficient condition for a non- singular ring to be reduced is given.
Highlights
Throughout this work all rings are assumed to be associative rings with identity
Recall that: 1- A ring R is said to be reduced if R contains no non- zero nilpotent element ; 2- A ring R is said to be a duo- ring if every right and left ideal is a two-sided ideal; 3- J(R) and N will stand respectively for the Jacobson radical ideal of R and the set of all nilpotent elements; 4The right and left annihilators of a in R will be denoted by r(a) and l(a), respectively ; 5- An ideal I is said to be a right pure if, for every a I, there exists b I such that a=ab(ba)
Definition 2.1: A non-zero elements a of R is said to be right singular if r(a) is an essential right ideal of R
Summary
Throughout this work all rings are assumed to be associative rings with identity. Recall that: 1- A ring R is said to be reduced if R contains no non- zero nilpotent element ; 2- A ring R is said to be a duo- ring if every right and left ideal is a two-sided ideal; 3- J(R) and N will stand respectively for the Jacobson radical ideal of R and the set of all nilpotent elements; 4The right and left annihilators of a in R will be denoted by r(a) and l(a), respectively ; 5- An ideal I is said to be a right (left) pure if , for every a I , there exists b I such that a=ab(ba).
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