Diagonability of idempotent matrices

A ring & (commutative with identity) with the property that every idempotent matrix over έ% is diagonable (i.e., similar to a diagonal matrix) will be called an ID-ring.We show that, in an ID-ring ^, if the elements a u α 2 , , a n e έ% generate the unit ideal then the vector [a lf a 2 , , α»] can be completed to an invertible matrix over &.We establish a canonical form (unique with respect to similarity) for the idempotent matrices over an ID-ring.We prove that if Λ^ is the ideal of nilpotents in & then & is an ID-ring if and only if ^\^yί^ is an ID-ring.The following are then shown to be ID-rings: elementary divisor rings, a restricted class of Hermite rings, 7r-regular rings, quasi-semi-local rings, polynomial rings in one variable over a principal ideal ring (zero divisors permitted), and polynomial rings in two variables over a π-regular ring with finitely many idempotents.