## 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].

### MSC:

 13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations 13D05 Homological dimension and commutative rings 13F20 Polynomial rings and ideals; rings of integer-valued polynomials

### Citations:

Zbl 0248.13001; Zbl 0861.13006