Correction to: “On ring structures determined by groups”

Abstract

PROOF. Lemma 2, we know that R=RTGRF, where RT is the torsion ideal of R and RF is the torsion-free ideal of R. Hence R/J -RT/JTERF/JF. But then RF/JF is a torsion which implies that RF = JF, since +RF is a torsion-free divisible group. Hopkin's Theorem [4 ] then forces RF=JF =O, and R is therefore a torsion In addition, Professor S. K. Jain has kindly made me aware of two errors of omission. The last line of Theorem 1 should read X 2 matrices over GF(2) or GF(3)., and lines 6 and 7 of the proof should read. . ... Furthermore, the 2 X 2 matrix rings over GF(2) and GF(3) are the only matrix rings with a solvable. The following should be added to the statement of Corollary 2 of Theorem 1: A of type (a) may occur as a direct summand with rings of types (b), (c), and (d), since its quasi-regular group has order 1. ADDED IN PROOF. The paragraph preceding Corollary 1 on p. 473 should now read: Since the quasi-regular group of the of 2X2 matrices over GF(2) or GF(3) is not nilpotent, we have the following corollary to Theorem 1. The line following the first paragraph on p. 474 should be deleted, while the first sentence in the proof of Theorem 2 (p. 475) should now read: By Propositions 2' and 3, R is a torsion ring. REMARK. P. B. Battacharya and S. K. Jain obtained essentially the same results in A note on the adjoint group of a ring (to appear in Archive der Mathematik) using different techniques. They extended these results in Rings having solvable adjoint groups (to appear in these Proceedings) where they prove: If R is a perfect and 'R is a finitely generated solvable group then R is finite, and and hence 'R = P, o P2 0 0 . . O Pm where the Pi are pairwise commuting p-groups. (Hyman Bass first defined perfect rings in Trans. Amer. Math. Soc. 95 (1960), 466-488.)