Ideals of distributive rings

A. A. Tuganbaev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 4 (1998) no. 2, pp. 791-794.

Résumé

Let $P$ be a prime ideal of a distributive ring $A$, and let $T$ be the set of all elements $t\in A$ such that $t+P$ is a regular element of the ring $A/P$. Then for any elements $a\in A$, $t\in T$ there exist elements $b_1,b_2\in A$, $u_1,u_2\in T$ such that $au_1=tb_1$, $u_2a=b_2t$. If either all square-zero elements of $A$ are central or $A$ satisfies the maximum conditions for right and left annihilators, then the classical two-sided localization $A_P$ exists and is a distributive ring.