Prime rings with involution whose symmetric zero-divisors are nilpotent

Abstract:

Let $k$ be a field and $R$ the $k$-algebra generated by $x$ and $y$ with the single defining relation ${x^2} = 0$. Using free ring techniques we prove that the set of left zero-divisors of $R$ is $Rx$. There is a unique involution fixing $x,y$ and this makes $R$ 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).