Prime rings with involution whose symmetric zero-divisors are nilpotent

Abstract

Let k k be a field and R R the k k -algebra generated by x x and y y with the single defining relation x 2 = 0 {x^2} = 0 . Using free ring techniques we prove that the set of left zero-divisors of R R is R x Rx . There is a unique involution fixing x , y x,y and this makes R R into a prime ring with involution whose symmetric zero-divisors are nilpotent (answering a question by W. S. Martindale). This example also provides us with a subfunctor of the identity whose value is a onesided ideal (answering a question by R. Baer).