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On continuous rings and self injective rings. (English) Zbl 0144.27301
Let $$S$$ be a ring with Jacobson radical $$N$$. The author’s main result is that if $$S$$ is left self-injective, then so is $$S/N$$. Moreover, if $$S$$ satisfies continuity conditions somewhat weaker than injectivity, an arbitrary set of orthogonal idempotents in $$S/N$$ lifts to a set of orthogonal idempotents in $$S$$. $$S$$ is called left continuous if any left ideal is an essential submodule of a left ideal $$Se$$ for some $$e = e^2\in S$$, and a left ideal isomorphic to $$Se$$ is also generated by an idempotent. Left self-injective rings are left continuous because of the existence of injective hulls. The author shows that for a left continuous ring $$S$$, $$S/N$$ is (von Neumann) regular, $$N$$ is the left singular ideal of $$S$$, and idempotents in $$S/N$$ lift to idempotents in $$S$$. These are well known properties of left self-injective rings (see the author [Proc. Japan Acad. 35, 16–21 (1959; Zbl 0085.25901)] and C. Faith and the author [Arch. Math. 15, 116–174 (1964; Zbl 0131.27502)]).
The key lemma states that for left continuous $$S$$, if $$e$$ and $$f$$ are idempotents in $$S$$ such that $$Se\cap Sf \subseteq N$$, then $$Se\cap Sf = 0$$. Thus idempotents independent in $$S/N$$ lift to idempotents independent in $$S$$. If $$S$$ is left continuous, this lemma plus the existence of essential extensions generated by idempotents enables the author to show any set of orthogonal idempotents in $$S/N$$ lift orthogonally, and if $$S$$ is left self-injective, maps from ideals of $$S/N$$ to $$S/N$$ lift to maps from $$S$$ to $$S$$ so $$S/N$$ is left self-injective.
The author also shows that, if $$S$$ is a direct sum of $$n$$ isomorphic left ideals for $$n > 1$$ (i. e. $$S = n\times n$$ matrices over some ring $$eSe)$$, then $$S$$ is left continuous if and only if $$S$$ is left injective.
If one defines left continuity of a module by replacing ideals generated by idempotents by direct summands in the definition, this is clearly a categorical property of a module. By the Morita category isomorphism (see K. Morita [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 6, 83–142 (1958; Zbl 0080.25702)]) if $$S$$ is left continuous, then $$S_n$$ is a sum of $$n$$ isomorphic left continuous ideals, so the author’s result shows that a direct sum of left continuous modules need not be left continuous. However, a direct sum of injective modules is injective, so $$S$$ is left self-injective if and only if $$S_n$$ is. The author shows this without resorting to the Morita category isomorphism.

##### MSC:
 16D50 Injective modules, self-injective associative rings
##### Keywords:
continuous rings; self-injective rings
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##### References:
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