RO-NMO 2016 Problem 122

Let be a ring $A,$ having the property that any element in the set $A\setminus U(A)$ is nilpotent. Show that

a) $axa=0, \forall x\in A, \forall a\in A\setminus U(A) .$ b) $\mathbb{N}\ni |A|\ge 2 \implies\left(\exists d\in A\setminus \left( U(A)\cup\{ 0 \}\right) d'\in A\setminus U(A)\implies dd'=d'd=0\right) .$

Ioan Băetu