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**Prüfer conditions in rings with zero-divisors.**
*(English)*
Zbl 1107.13023

Chapman, Scott T. (ed.), Arithmetical properties of commutative rings and monoids. Boca Raton, FL: Chapman & Hall/CRC (ISBN 0-8247-2327-9/pbk). Lecture Notes in Pure and Applied Mathematics 241, 272-281 (2005).

A Prüfer ring is a commutative ring such that every finitely generated regular ideal is invertible. A great number of equivalent conditions characterizes Prüfer domains [see R. Gilmer, Multiplicative ideal theory. Pure and Applied Mathematics. Vol. 12 (1972; Zbl 0248.13001) and M. Fontana, J. A. Huckaba and I.J. Papick, Prüfer domains. Pure and Applied Mathematics, Marcel Dekker. 203 (1997; Zbl 0861.13006)].

In this nice paper, after giving a short historical survey of Prüfer domains, four equivalent characterizations of a Prüfer domain are extended and studied in the context of rings with zero divisors. Using some known and new results, the author shows the following implications:

Let \(R\) be a ring with zero divisors. Then, \(R\) is a semihereditary ring \(\Rightarrow\) w.dim \(R\leq 1\Rightarrow R\) is an arithmetical ring \(\Rightarrow R\) is a Gaussian ring \(\Rightarrow R\) is a Prüfer ring.

Although all these implications are reversible in case of an integral domain, for each of them, a counterexample is given of a ring with zero divisors which does not satisfy the reverse implication. Moreover, each of the three first implications has a converse if an additional assumption is considered. More precisely, the first implication is an equivalence when \(R\) is a coherent ring. When \(R\) is reduced, we have w.dim \(R\leq 1\Leftrightarrow R\) is an arithmetical ring \(\Leftrightarrow R\) is a Gaussian ring and when \(R\) is a PP ring or when its total quotient ring of quotients is a von Neumann regular ring, then \(R\) is a semihereditary ring \(\Leftrightarrow\) w.dim \(R\leq 1\Leftrightarrow R\) is an arithmetical ring \(\Leftrightarrow R\) is a Gaussian ring.

For the entire collection see [Zbl 1061.13001].

In this nice paper, after giving a short historical survey of Prüfer domains, four equivalent characterizations of a Prüfer domain are extended and studied in the context of rings with zero divisors. Using some known and new results, the author shows the following implications:

Let \(R\) be a ring with zero divisors. Then, \(R\) is a semihereditary ring \(\Rightarrow\) w.dim \(R\leq 1\Rightarrow R\) is an arithmetical ring \(\Rightarrow R\) is a Gaussian ring \(\Rightarrow R\) is a Prüfer ring.

Although all these implications are reversible in case of an integral domain, for each of them, a counterexample is given of a ring with zero divisors which does not satisfy the reverse implication. Moreover, each of the three first implications has a converse if an additional assumption is considered. More precisely, the first implication is an equivalence when \(R\) is a coherent ring. When \(R\) is reduced, we have w.dim \(R\leq 1\Leftrightarrow R\) is an arithmetical ring \(\Leftrightarrow R\) is a Gaussian ring and when \(R\) is a PP ring or when its total quotient ring of quotients is a von Neumann regular ring, then \(R\) is a semihereditary ring \(\Leftrightarrow\) w.dim \(R\leq 1\Leftrightarrow R\) is an arithmetical ring \(\Leftrightarrow R\) is a Gaussian ring.

For the entire collection see [Zbl 1061.13001].

Reviewer: Martine Picavet-L’Hermitte (Le Cendre)