Multiplicative sets of idempotents in a finite ring

Let R be a finite ring with identity, T its set of idempotents. We study the subsets of T that can be closed under multiplication and the implications that fact has to the structure of R. We consider the subset M of all minimal idempotents and zero and prove that M is closed under multiplication if and only if R is a direct sum of local rings. We achieve this by studying the properties of units that are preserved by idempotents.