A connection between weak regularity and the simplicity of prime factor rings

Abstract

In this paper, we show that a reduced ring <italic>R</italic> is weakly regular (i.e., <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I squared equals upper I"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{I^2} = I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each one-sided ideal <italic>I</italic> of <italic>R</italic>) if and only if every prime ideal is maximal. This result extends several well-known results. Moreover, we provide examples which indicate that further generalization of this result is limited.