Diagonability of idempotent matrices

A ring & (commutative with identity) with the property that every idempotent matrix over % 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 2 , , a n e % generate the unit ideal then the vector [a lf a 2 , , ] can be completed to an invertible matrix over &.