Abstract
Let R be a ring whose set of idempotents E(R) is closed under multiplication. When R has an identity 1, E(R) is known to lie in the center of R, thus forming a Boolean algebra. In this article we consider what occurs if R has no identity, in which case E(R) is a possibly noncommutative variant of a generalized Boolean algebra. We explore the effects of E(R) on the structure of R, giving attention to various induced decompositions of R.
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