Hereditary rings integral over their centers (Q1072620)

Throughout this review R will denote a right hereditary ring with centre C. It was shown by Bergman that C need not be hereditary, and indeed C need not be hereditary even when R is also a right Noetherian P.I. ring [\textit{L. W. Small} and \textit{A. R. Wadsworth}, Commun. Algebra 9, 1105- 1118 (1981; Zbl 0453.16008)]. On the other hand certain extra conditions on R do force C to be hereditary, e.g. when R is a finite C-module [\textit{S. Jøndrup}, J. Lond. Math. Soc., II. Ser. 15, 211-212 (1977; Zbl 0353.16012)]. In the present paper it is shown that if R is integral over C then C is hereditary. It is also shown that if (1) R is integral over C, (2) the non-zero elements of C are not zero-divisors in R, and (3) C is not a field, then R is a direct sum of left and right Noetherian hereditary prime rings. In particular, if R is prime right hereditary integral over its centre C and C is not a field then R is left and right Noetherian, but if C is a field then an example shows that R need not be Noetherian.