Let R, and R, be algebras with 1 over a common field F, with each (Rj: F) > 1, and let R = R, II R, denote the coproduct (often called the free product) of R, and R, over F. It is easy to show that R is always prime; in fact R is primitive in case at least one of (Ri: F) > 2 171. On the other hand, when (Ri: F) = 2, i= 1,2, it is clear from [ 1, p. 261 that R is prime PI but never primitive. If R, and R, are domains then we have Cohn’s result [2 i that R is again a domain. In studying certain automorphisms of a prime ring R one is led in a natural way to the normal closure of R, and we now briefly recall the meaning of this notion. If R, is the left quotient ring of R relative to the filter F of all nonzero two-sided ideals of R, the set N* of all units u of R-,such that uwlRu = R is called the set of normalizing elements [ 10, p. 5 ]. The automorphisms thus induced on R are just the X-inner automorphisms of Kharchenko [IO, p. 31. The subring RN of R generated by R and N = N* U (0) is called the normal closure of R. An important subset of N is the center C of R-, (called the extended center of R); it is known that C is a field. We now return to the coproduct R = R, II R, , where R L and R2 are arbitrary F-algebras. It is known in this situation that if at least one of (Ri: F) > 2, then C = F [8]; when both (R,: F) = 2, then C =F(tj G R [ 1 j. Now fix F-bases {xii u 1 for R, and ( ~7~) u 1 for R2 and call the various products of alternating xI(s and -yj’s basis monomials. The degree of a basis
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