Vol. 19, No. 3, 1966

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
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