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On an intrinsic definition of weakly normal rings. (English) Zbl 0586.13006

Originally a commutative ring A was defined to be weakly normal (seminormal) if A was equal to its weak normalization (seminormalization). R. G. Swan [J. Algebra 67, 210-219 (1980; Zbl 0473.13001)] defined A to be seminormal if A is reduced and for any elements b, c in A with \(b^ 3=c^ 2\), there is an element a in A with \(a^ 2=b\) and \(a^ 3=c\). He showed that this definition agrees with the classical definition and gave a thorough investigation of seminormality. The purpose of this paper is to give a treatment of weakly normal rings along the lines of Swan’s treatment of seminormal rings. The intrinsic definition of weakly normal rings mentioned in the title is that A is weakly normal if A is seminormal and for any elements b, c, d, e in A with d a nonzero divisor with \(c^ p=bd^ p\) and \(pc=de\) for some prime p, there is an element a in A with \(b=a^ p\) and \(e=pa\).
Reviewer: D.D.Anderson

MSC:

13A99 General commutative ring theory
13B02 Extension theory of commutative rings

Citations:

Zbl 0473.13001
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