The fineness properties of Morita contexts

Abstract

Let [Formula: see text] be a Morita context. For generalized fine (respectively, generalized unit-fine) rings [Formula: see text] and [Formula: see text], it is proved that [Formula: see text] is generalized fine (respectively, generalized unit-fine) if and only if, for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] and [Formula: see text] implies [Formula: see text]. Especially, for fine (respectively, unit-fine) rings [Formula: see text] and [Formula: see text], [Formula: see text] is fine (respectively, unit-fine) if and only if, for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] and [Formula: see text] implies [Formula: see text]. As consequences, (1) matrix rings over fine (respectively, unit-fine, generalized fine and generalized unit-fine) rings are fine (respectively, unit-fine, generalized fine and generalized unit-fine); (2) a sufficient condition for a simple ring to be fine (respectively, unit-fine) is obtained: a simple ring [Formula: see text] is fine (respectively, unit-fine) if both [Formula: see text] and [Formula: see text] are fine (respectively, unit-fine) for some [Formula: see text]; and (3) a question of Cǎlugǎreanu [1] on unit-fine matrix rings is affirmatively answered.