Rings whose idempotents form a multiplicative set Karin Cvetko-Vah and Jonathan Leech May 13, 2009 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; moreover each idempotent e induces a decomposition eR ⊕ (1 - e)R of R. In this paper 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 R with attention given to the decompositions of R induced from decompositions of E(R) as well as to the indecomposable cases. Keywords: idempotent element, normal band, D-class, skew Boolean algebra. Introduction A standard exercise in elementary ring theory states that given a ring with identity 1 whose idempotents are closed under multiplication, all the idempotents lie in the center of the ring and thus form a Boolean algebra with operations: e ∧ f = ef , e ∨ f = e + f − ef (i.e., the circle operation e ◦ f ) and e ′ =1 − e. What happens when E(R) is a multiplicative in a ring R without identity? In this case E(R) is immediately seen to be a normal band. That is, E(R) is a band (a semigroup of idempotents) on which xyzw = xzyw holds. For some time it has been known that any band S of idempotents in a ring that is maximal with respect to being normal is likewise closed under a counter-product, e∇f =(e ◦ f ) 2 , that is also associative and idempotent. In this case S forms a noncommutative variant of a Boolean algebra called a skew Boolean algebra. Its meet is again multiplication, its 1