Multiplicative classification of associative rings

A. V. Mikhalev

Sbornik. Mathematics, Tome 63 (1989) no. 1, pp. 205-218

Résumé

Let $R$ be a ring, $l(a)$ and $r(a)$ the left and right annihilators of the element $a\in R$, $\mathrm{AC}(R)=\sum_{a,b\in R}l(a)bl(b)a$ the two-sided ideal in $R$ called the additive controller, and let $\alpha\colon R\to S$ be an $m$-isomorphism (i.e., multiplicative isomorphism) and $D(\alpha)=\{[(x+y)^\alpha-x^\alpha-y^\alpha]^{\alpha^{-1}}/x,y\in R\}$ its defect. An ideal $I$ in the ring $R$ is called an $m$-ideal if for all $m$-isomorphisms $\alpha\colon R\to S$, $L^\alpha$ is an ideal in $S$ and $a-b\in L$ if and only if $a^\alpha-b^\alpha\in L^\alpha$. It is shown that $$ D(\alpha)\mathrm{AC}(R)=0=\mathrm{AC}(R)D(\alpha). $$ Very general sufficient conditions are given that a multiplicative isomorphism of subsemigroups of multiplicative semigroups of rings be extendible to the isomorphism of the subrings generated by them. Minimal prime ideals and the prime radical of a ring are $m$-ideals. The strongly regular and regular rings that have unique addition are characterized. Bibliography: 29 titles.