Math 6310 • Algebra

Let \(R\) be a semisimple right Artinian ring (in particular, this implies that \(R\) has an identity 1). Assume \(x\in R\) and define the sets \(A_x\) and \(B_x\) by \[ A_x=\{y\in R\mid y\ne 0\text { and }xy=0\} \quad \text {and}\quad B_x=\{y\in R\mid xy=1\}. \] Prove that exactly one of these sets is empty.