Abstarct: In this thesis we prove that if R has a complete finite set of primitive orthogonal idempotents, then R is a finite direct product of connected rings precisely when M(R) is multiplicative. We prove that if R is a (von Neumann) regular ring with M(R) multiplicative, then every primitive idempotent in R is central. It is also shown that this does not happen even in semihereditary and semiregular rings. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates. The central idempotents of any ring with identity form a Boolean algebra. This result is largely extended for rings with generalized commuting idempotents. We show that I(R) is a finite additive set if and only if M(R) /{0} is a complete set of primitive central idempotents, char(R) = 2 and every nonzero idempotent of R can be expressed as a sum of orthogonal primitive idempotents of R. Also we show that for a regular ring R such that I(R) is a finite additive set, if the multiplicative group of all units of R is abelian , then R is a commutative ring. For idempotent e, it is shown that e ϵ Sl (R) if and only if re = ere for all nilpotent elements r є R if and only if fe ϵ I(R) for all f є I(R) if and only if fe = efe for all f є I(R) if and only if fe = efe for all f є I(R) which are isomorphic to e. For a ring R having a complete set of centrally primitive idempotents, we show that every nonzero left semicentral idempotent is a finite sum of orthogonal left semicentral primitive idempotents, and eRe has also a complete set of primitive idempotents for any 0≠e є Sl(R).