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Hereditary rings integral over their centers. (English) Zbl 0587.16003

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 [L. W. Small and 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 [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.
Reviewer: A.W.Chatters

MSC:

16N60 Prime and semiprime associative rings
16U30 Divisibility, noncommutative UFDs
16S20 Centralizing and normalizing extensions
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
16P40 Noetherian rings and modules (associative rings and algebras)
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