Rings with Multiplicative Sets of Primitive Idempotents

Abstract

Let R be a ring and M(R) the set consisting of zero and primitive idempotents of R. We study the rings R for which M(R) is multiplicative. It is proved 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. Let R be an arbitrary ring with M(R) multiplicative and e ∈ R be a primitive idempotent, then for every unit u ∈ R, it is proved that eue is a unit in eRe. We also prove that if M(R) is multiplicative, then two primitive idempotents e and f in R are conjugates, i.e., f = ueu ⁻¹ for some u ∈ U(R), if and only if ef ≠ 0.

Key Words:

• Central idempotents
• Connected rings
• Primitive idempotents
• von Neumann regular rings

2000 Mathematics Subject Classification:

• Primary 16U99
• Secondary 16D70, 16E50

ACKNOWLEDGMENT

The Research of the first author is supported by CSIR, India, and will form a part of her Ph.D dissertation.

Notes

Communicated by E. R. Puczylowski.