A ring ℛ (commutative with
identity) with the property that every idempotent matrix over ℛ is diagonable (i.e.,
similar to a diagonal matrix) will be called an ID-ring. We show that, in an
ID-ring ℛ, if the elements a1,a2,⋯ , an∈ℛ generate the unit ideal then the
vector [a1,a2,⋯,an] can be completed to an invertible matrix over ℛ. We
establish a canonical form (unique with respect to similarity) for the idempotent
matrices over an ID-ring. We prove that if 𝒩 is the ideal of nilpotents in
ℛ then ℛ is an ID-ring if and only if ℛ∕𝒩 is an ID-ring. The following
are then shown to be ID-rings: elementary divisor rings, a restricted class
of Hermite rings, π-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.
