ADDITIVE SET OF IDEMPOTENTS IN RINGS

Let R be a ring with identity 1, I(R) be the set of all nonunit idempotents in R, and M(R) be the set of all primitive idempotents and 0 of R. We say that I(R) is additive if for all e, f is an element of I(R) (e not equal f), e + f is an element of I(R), and M(R) is additive in I(R) if for all e, f is an element of M(R) (e not equal f), e + f is an element of I(R). In this article, the following points are shown: (1) I(R) is additive if and only if I(R) is multiplicative and the characteristic of R is 2; M(R) is additive in I(R) if and only if M(R) is orthogonal. If 0 not equal ef is an element of I(R) for some e is an element of M(R) and f is an element of I(R), then ef is an element of M(R), (2) If R has a complete set of primitive idempotents, then R is a finite product of connected rings if and only if I(R) is multiplicative if and only if M(R)is additive in I(R).