Rigid Artinian rings

Abstract

In [4], Maxson studied the properties of a ring R whose only ring endomorphisms φ: R → R are the trivial ones, namely the identity map, idR, and the map 0R given by φ(R) = 0. We shall say that any such ring is rigid, slightly extending the definition used in [4] by dropping the restriction that R2 ≠ 0. Maxson's most detailed results concerned the structure of rigid artinian rings, and our main aim is to complete this part of his investigation by establishing the followingTheorem. Let R(≠0) be a left-artinian ring. Then R is rigid if and only if(i) , the ring of integers modulo a prime power pk,(ii) R ≅ N2, the null ring on a cyclic group of order 2, or(iii) R is a rigid field of characteristic zero.