Divinsky’s radical

Abstract:

Let $F$ and $R$ be rings, $M$ an $F - R$-bimodule, and $\Delta$ the largest $F$-submodule $N$ of $M$ such that for each $x \in N,fx = x$ for some $f \in F$. (1) If either $F$ or $M$ satisfies the minimum condition then $\Delta = {F^k}M$ for some positive integer $k$; provided that whenever $x \in {F^\omega }M = \cap _{n = 1}^\infty ({F^n}M)$ and $Fx \subseteq \Delta$, then $x \in \Delta$. (2) If $M$ satisfies the maximum condition and $F = ({f_1}, \cdots ,{f_n})$ where ${f_1}, \cdots ,{f_n}$ is a normalising set of generators (that is, \[ {f_i}F = F{f_i}\operatorname {modulo} ({f_1}, \cdots ,{f_{i - 1}})\] for each $i = 1, \cdots ,n)$, then $\Delta = {F^\omega }M$. (3) If $M = F = R,\Delta = (0)$, $R$ satisfies the maximum condition, and $R$ has a normalising set of generators, then $R$ can be embedded in a Jacobson radical ring.