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An example is produced of an algebra, embedded in 2 x 2 matrices over a field, which is not PI when an element's inverse is formally adjoined. This example is used to show that the generic 2 x 2 matrices, a domain, has the same property. In commutative algebra the procedure of adjoining the inverse of an element in a ring has proven useful. Recently work has been done towards generalizing the concepts of commutative algebra and algebraic geometry to PI-algebras [1], [2], [6]. In this paper it is shown that the algebra which is obtained from a subring of a ring of 2 x 2 matrices over an extension of the base field, by formal adjunction of the inverse of an element is not PI. We do this by describing an irreducible infinite dimensional representation and applying Kaplansky's theorem [4]. For A and B algebras over a commutative ring C, we shall use the notation of the free product A *c B as found in [5]. This C-algebra is the coproduct in the category of C-algebras. For R an algebra over a field k, and x an element of R, let x' denote the residue class of z in the algebra R *k k[z]/(zx 1, xz 1). Define R{x'} to be the above algebra. Let k{X, Y} be the algebra obtained by adjoining two generic 2 x 2 matrices to a field k. THEOREM 1. k{X, Y} {X'} is not a PI-algebra. To prove this theorem we shall work with another algebra. We define A to be the quotient of the free algebra k{x, y} by the relations y2 = 0 and yx'y = 0, for i > 0. LEMMA 2. The set S = {xiyxJ, xk; ij,k > 0) spans A as a vector space over k. PROOF. The set S u {0} is closed under multiplication, since x vxJxkyxt = 0, for ij,k,n > 0. Thus since A is generated as an algebra by x andy, the ring A is spanned by S over k. Note. S is actually a basis for A. Received by the editors July 25, 1978 and, in revised form, October 18, 1978. AMS (MOS) subject classifications (1970). Primary 16A08; Secondary 16A38.

A PI-algebra which is not PI when an inverse is adjoined