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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Divinsky's radical


Author: Patrick N. Stewart
Journal: Proc. Amer. Math. Soc. 31 (1972), 347-353
MSC: Primary 16.30
DOI: https://doi.org/10.1090/S0002-9939-1972-0286823-3
MathSciNet review: 0286823
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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,

$\displaystyle {f_i}F = F{f_i}\operatorname{modulo} ({f_1}, \cdots ,{f_{i - 1}})$

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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0286823-3
Keywords: Divinsky's radical, $ D$-regularity, embed in a Jacobson radical ring, intersection theorem, AR property, quasi quotient ring
Article copyright: © Copyright 1972 American Mathematical Society