A note on rigid rings

Let R be an associative ring with identity element and let sigma be an endomorphism of R. sigma is called rigid if whenever a is an element of R with a sigma(a) = 0, then a = 0. R is called a sigma-rigid ring if it has a rigid endomorphism sigma. In this article we study some of the elementary properties of rigid rings which are generalizations of some properties of reduced rings. We show, among other things, that the set of all rigid automorphisms of a ring R is a normal subgroup of all automorphisms of R.