Vol. 19, No. 3, 1966

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ISSN: 0030-8730
Diagonability of idempotent matrices

Arthur Steger

Vol. 19 (1966), No. 3, 535–542

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 a1,a2, , an ∈ℛ generate the unit ideal then the vector [a1,a2,,an] 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 𝒩 is the ideal of nilpotents in then is an ID-ring if and only if 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.

Mathematical Subject Classification
Primary: 13.93
Secondary: 16.48
Received: 8 July 1965
Published: 1 December 1966
Arthur Steger