Involutions in left Artinian rings

The main work of this paper is to investigate left Artinian rings with identity having finitely many involutions. An element a in a ring A is an involution if {dollar}rm asp2{dollar} = 1. The new results are as follows: (1) Left Artinian rings A with identity having only one or two involutions are characterized. The left Artinian rings for which 2 is a unit in A and in which the set of involutions in A forms a finite abelian group are also characterized. (2) Let A be a ring of all n x n matrices over a division ring. If a is a nilpotent element of A, then 1 + a is the product of two involutions. As a corollary, we establish that the above result also holds in any semisimple left Artinian ring and in the upper triangular matrix ring over a division ring. (3) The invertible diagonal matrices over the ring of real quaternions are investigated and the conditions for them to be a product of finite involutions are obtained.