Orders in Uniserial Rings

A. A. Tuganbaev

Matematičeskie zametki, Tome 74 (2003) no. 6, pp. 924-933

Résumé

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 $b_1\in A$ and $t_1\in T(A)$ with $a_1b_1 = a_2t_1$ or there exist two elements $b_2\in A$ and $t_2\in T(A)$ with $a_2b_2 = a_1t_2$. 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.