On the irreducibility of slice regular algebraic sets

In the present paper we investigate the relations between irreducible slice regular algebraic sets in and quasi prime right ideals of the ring of slice regular polynomials in n quaternionic variables. We provide algebraic conditions on right ideals of slice regular polynomials which guarantee the irreducibility of the corresponding commuting slice regular algebraic sets and show that radical right ideals associated with irreducible commuting slice regular algebraic sets are quasi prime. Furthermore we establish that this correspondence is an equivalence in the case of principal right ideals.