Powerful ideals, strongly primary ideals, almost pseudo-valuation domains, and conducive domains. (English) Zbl 1063.13017

Badawi, Ayman; Houston, Evan

Commun. Algebra 30, No. 4, 1591-1606 (2002).

Summary: Let \(R\) be a domain with quotient field \(K\), and let \(I\) be an ideal of \(R\). We say that \(I\) is powerful (strongly primary) if whenever \(x,y\in K\) and \(xy\in I\), we have \(x\in R\) or \(y\in R\) \((x\in I\) or \(y^n\in I\) for some \(n\geq 1)\). We show that an ideal with either of these properties is comparable to every prime ideal of \(R\), that an ideal is strongly primary \(\Leftrightarrow\) it is a primary ideal in some valuation overring of \(R\), and that \(R\) admits a powerful ideal \(\Leftrightarrow R\) admits a strongly primary ideal \(\Leftrightarrow R\) is conducive in the sense of D. E. Dobbs and R. Fedder [J. Algebra 86, 494–510 (1984; Zbl 0531.13002)]. Finally, we study domains each of whose prime ideals is strongly primary.

MSC:

13G05	Integral domains
13A15	Ideals and multiplicative ideal theory in commutative rings

Keywords:

conducive domains; powerful ideal; almost pseudo-valuation domains

Citations:

Zbl 0531.13002

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