The fineness properties of Morita contexts

摘要

Let R = AMNB be a Morita context. For generalized fine (respectively, generalized unit-fine) rings A and B, it is proved that R is generalized fine (respectively, generalized unit-fine) if and only if, for a∈A and b∈B, MbN⊆J(A) implies b∈J(B) and NaM⊆J(B) implies a∈J(A). Especially, for fine (respectively, unit-fine) rings A and B, R is fine (respectively, unit-fine) if and only if, for a∈A and b∈B, MbN=0 implies b=0 and NaM=0 implies a=0. 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 R is fine (respectively, unit-fine) if both eRe and (1−e)R(1−e) are fine (respectively, unit-fine) for some e2=e∈R; and (3) a question of Cǎlugǎreanu [1] on unit-fine matrix rings is affirmatively answered.