Idempotents in matrix rings

Abstract:

Let R be a commutative, von Neumann regular ring and ${M_n}(R)$ the ring of $n \times n$ matrices over R. What are the idempotents in ${M_n}(R)$ ? Our motivation is to think of R as the sort of ring that occurs in functional analysis, for example a ring of measurable functions. We show how to uniquely write down all idempotents in ${M_n}(R)$ in terms of arbitrary parameters. The main theorem is stated in language to appeal to an audience wider than algebraists, but in a remark, we give a more refined statement for specialists.