On continuous rings and self injective rings.

*(English)*Zbl 0144.27301Let \(S\) 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.

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 |

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##### References:

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