Abstract

A ring <TEX>$R$</TEX> is called semiregular if <TEX>$R/J$</TEX> is regular and idem-potents lift modulo <TEX>$J$</TEX>, where <TEX>$J$</TEX> denotes the Jacobson radical of <TEX>$R$</TEX>. We give some characterizations of rings <TEX>$R$</TEX> such that idempotents lift modulo <TEX>$J$</TEX>, and <TEX>$R/J$</TEX> satisfies one of the following conditions: (one-sided) unit-regular, strongly regular, (unit, strongly, weakly) <TEX>${\pi}$</TEX>-regular.

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