Extensions of Jacobson's lemma

GUSTAVO CORACH, BHAGGY DUGGAL, ROBIN HARTE

COMMUNICATIONS IN ALGEBRA

TAYLOR & FRANCIS INC

Lugar: Londres; Año: 2013 vol. 41 p. 520 - 531

0092-7872

Jacobson’s Lemma says that if a
c ∈ A thenac − 1 ∈ A−1 ⇐⇒ ca − 1 ∈ A−1 which holds separately for the left and the right invertibles of A, as well asfor the non zero-divisors of A. In this note, we generalize the identity above and many of its relativesfrom ca − 1 to certain ba − 1: specifically we will supposeaba = aca.Three special cases are of interest: the caseb = c
which will give Jacobson’s lemma; the case in whichaba = aca = a
in which both b and c are generalized inverses of a ∈ A; and the caseaba = a^2
in which c = 1. This last case goes back to Vidav; in particular,Schmoeger shows that aba=a^2 holds if there are idempotents p = p^2
q = q^2 forwhich a = qp
and b = pq.The central results in this note are of course pure algebra: but in theneighboring realm of topological algebra they have very close relatives, and we takethe opportunity to extend our purely algebraic observations to their topologicalanalogues.