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Almost maximal Prüfer domains. (English) Zbl 0977.13010

Extending a definition of I. Kaplansky, W. Brandal [Trans. Am. Math. Soc. 183, 203-222 (1973; Zbl 0273.13011)] defined a commutative ring \(R\) to be almost maximal if every proper homomorphic image of \(R\) is linearly compact. He proved that if \(D\) is an almost maximal Prüfer domain with quotient field \(K\), then every homomorphic image of \(D\) is an injective \(D\)-module.
In the paper under review, the converse of Brandal’s theorem is established. It is also shown that a Prüfer domain \(D\) is almost maximal if and only if each nonzero prime ideal of \(D\) has injective dimension one. Applications involving reflexive domains are also considered.

MSC:

13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13C11 Injective and flat modules and ideals in commutative rings
18G05 Projectives and injectives (category-theoretic aspects)
13J99 Topological rings and modules

Citations:

Zbl 0273.13011
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References:

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