Additive Set of Idempotents in Rings

Abstract

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 ∈ I(R) (e ≠ f), e + f ∈ I(R), and M(R) is additive in I(R) if for all e, f ∈ M(R)(e ≠ f), e + f ∈ 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 ≠ ef ∈ I(R) for some e ∈ M(R) and f ∈ I(R), then ef ∈ 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).

Key Words:

• Connected ring
• Fully basic ring
• Primitive idempotents

2010 Mathematics Subject Classification:

• Primary 16U99
• Secondary 06E20

ACKNOWLEDGMENTS

The authors would like to thank the referee for his/her careful checking of the details and helpful comments for making the article more readable.

This work was supported by research funds from Dong-A University.

Notes

Communicated T. Albu.