Rings in which nilpotents form a subring

Abstract

Let R be a ring with the set of nilpotents Nil(R). We prove that the following are equivalent: (i) Nil(R) is additively closed, (ii) Nil(R) is multiplicatively closed and R satisfies Köthe’s conjecture, (iii) Nil(R) is closed under the operation x ◦ y = x + y − xy, (iv) Nil(R) is a subring of R. Some applications and examples of rings with this property are given, with an emphasis on certain classes of exchange and clean rings.