Orders in Uniserial Rings

Abstract

Let A be a ring, and let T(A) and N(A) be the set of all the regular elements of A and the set of all nonregular elements of A, respectively. It is proved that A is a right order in a right uniserial ring if and only if the set of all regular elements of A is a left ideal in the multiplicative semigroup A and for any two elements a1 and a2 of A, either there exist two elements b1 ∈ A and t1 ∈ T(A) with a1b1 = a2t1 or there exist two elements b2∈ A and t2∈ T(A) with a2b2 = a1t2. A right distributive ring A is a right order in a right uniserial ring if and only if the set N(A) is a left ideal of A. If A is a right distributive ring such that all its right divisors of zero are contained in the Jacobson radical J(A) of A, then A is a right order in a right uniserial ring.