PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 85, Number 3, July 1982 ON A THEOREM OF FLANDERS ROBERT E. HARTWIG ABSTRACT. It is shown that if Ft is a regular strongly-pi-regular ring, then R is unit-regular precisely when (ab)d s» (àa)d for all o, b £ Ft. This generalizes a result by Flanders, which states that the matrices AB and BA over a field F have the same elementary divisors except possibly those divisible by X. 1. Introduction. A classic theorem of Flanders [3] states that if A and B are nXn matrices over a field F, then AB and BA have the same elementary divisors, except possibly for those that are powers of X. The purpose of this note is to point out that the real reason why this result is true, is because the matrix ring FnXn is both strongly-pi-regular as well as unit- regular. We shall use the concept of pseudo-similarity, introduced in [5] to provide the necessary link between strong-pi-regularity and unit-regularity. We recall that a ring R is called (unit) regular, if for every a G R, there exists a (unit) solution x G R, to the equation axa = a. Such solutions will be denoted at a~. A ring R is strongly-pi-regular, S7rr for short, if for every a G R, there is a solution to the equations akxa = ak, xax = x, ax = xa, for some fc > 0. The solution is unique and is called the Drazin inverse ad of a [2]. In the special cases where fc = 0 or fc = 1, ad is called the group inverse of o, and is denoted by a*. It is well known that the ring FnXn of n X n matrices over a field F is both S7rr and unit-regular. Two ring elements are called pseudo-similar if x~ ax = b, xbx~ = a, xx~x = x for some x,x~ G R- It was shown in [6], that for a unit-regular ring, similarity («) and pseudo-similarity (~), coincide. Two idempotents e and / in R, are said to be equivalent, e ~ /, if eR and fR are isomorphic ( = ) as B-modules. This may be rewritten as e ~ / if e = p+p, f = PP+, for some p, p+ G R, that satisfy pp+p = p and p+pp'T' = p+. It is easily seen that e ~ / <=► e ~ / [4] and that ~ actually coincides with the classical P-relation on semigroups. 2. Main results. Our generalization of the theorem of Flanders is based on the following two main results. THEOREM 1. Let R be a strongly-pi-regular ring with unity and let x,y G R- Then the following are equivalent. (i) xd « yd, (ii) x2xd « t/V • In which case Received by the editors July 10, 1980 and, in revised form, December 9, 1981. 1980 Mathematics Subject Classification. Primary 15A21, 15A09. © 1982 American Mathematical Society 0002-9939/81/0000-0789/S01.75 310 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use