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 a ₁ and a ₂ of A, either there exist two elements b ₁ ∈ A and t ₁ ∈ T(A) with a1b₁ = a ₂t₁ or there exist two elements b ₂∈ A and t ₂∈ T(A) with a ₂ b ₂ = a ₁ t ₂. 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.