Crossed products of simple rings

Abstract

Let A be the algebra of all q X q matrices over division ring D. Suppose there is given group G of n automorphisms of A such that (i) the fixed subring S of G is simple ring such that [A: S] = n and (ii) GA, coincides with the totality of all homomorphisms of the S-left module A to itself where Ar is the ring of right multiplication of elements of A. Suppose also that there is given factor system {ca} (o-, r&G) in the center K of A. Then crossed product of A and G is defined via the same formulae as in the commutative case. (See [2 ].) The purpose of this note is to investigate the splitting property of factor system by an extension of S as well as of A. This is generalization of wellknown theorem for the commutative case as well as of result given in [2]. To begin with, we shall consider purely transcendental extension of A as follows. Let xi, * * *, Xm be m variables. Let D [xl, . . ., xm] be the polynomial ring of xi, * * *, Xm over D. We suppose xi lie in the center of the ring. Then the quotient division ring of D [xi, ..., xm ] is denoted by D (xi, * * * , xm). The existence of the quotient ring is clear from general theory of quotient ring, or it can be proved directly as follows. Generally let r be ring with no divisor of zero. Moreover suppose that for any nonzero elements and of r there exist nonzero elements a', b', a and b such that aa' = bb' and aa=bb. We consider the set of formal elements a-lb and cd-1 (a, b, c, dEr and #5O, d # O). Define aT lb1 = aj 'b2 if and only if there exist nonzero elements c and d such that aic =bid and a2c= b2d. It is good exercise to show that the above equivalent relation is well defined. Similarly, define biaT' = b2a' 1 if and only if there exist nonzero elements c and d such that ca1 = db1 and ca2=db2. Also define a-lb = cd-1 if and only if ac = bd. To verify that this is well defined is also good exercise. How to define the algebraic operations in this set is now almost clear. For example, (a-lb) (c-ld) = (c'a)-1b'd, where bc= c'-lb'. Also, a-lb +c-'d = (ca)'(cb + ad), where ac-' = c`-la. The set is division ring called quotient ring of r and it is uniquely determined up to isomorphism. To apply this general theory to our case, we must verify that the above mentioned conditions of r are satisfied for D [Xi, . . . , Xm]. To see it, take D [xi] first. The above