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A connection between weak regularity and the simplicity of prime factor rings. (English) Zbl 0814.16001

An associative ring \(R\) with identity is called reduced if it does not have non-zero nilpotent elements. The authors prove that a reduced ring \(R\) is weakly regular (i.e., \(I^ 2 = I\) for each one-sided ideal \(I\) of \(R\)) if and only if each prime ideal is maximal. The paper ends with two examples which indicate that further generalization is limited. Recently, K. Bejdar and R. Wisbauer announced a generalization [in Usp. Mat. Nauk 48, 161-162 (1993; Zbl 0811.16021)]; and E. P. Armendariz also announced a generalization [in Abstr. Am. Math. Soc. 14, 732 (1993)].

MSC:

16D30 Infinite-dimensional simple rings (except as in 16Kxx)
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16N60 Prime and semiprime associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras

Citations:

Zbl 0811.16021
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References:

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