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Two results concerning the ring R generated by a pair of 2 x 2 generic matrices over a field K are proved: (1) The trace ring of R is a coproduct of commutative rings. (2) If a finite subgroup G of SL(2, K) acts homogeneously on R and the characteristic of K does not divide the order of G, then the fixed ring RG is a finitely generated if-algebra.

Groups acting on the ring of two 2×2 generic matrices and a coproduct decomposition of its trace ring