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Well-behaved prime \(t\)-ideals and almost Krull domains. (English) Zbl 1506.13006

Let \(R\) be a domain. In [M. Zafrullah, J. Pure Appl. Algebra 65, No. 2, 199–207 (1990; Zbl 0705.13001)] the author called a prime \(t\)-ideal of \(R\) is well-behaved if \(PR_P\) is a \(t\)-ideal in \(R_P\). It is easy to see that if \(P\) is well behaved in \(R\), then \(PR_S\) is a \(t\)-ideal in \(R_S\) for each multiplicatively closed subset of \(R\) disjoint from \(P\). In this paper, the authors introduce the concept of well-behaved of a \(t\)-ideal \(I\) of \(R\) and study its properties. Call a \(t\)-ideal \(I\) of an integral domain \(R\) well-behaved if \(IR_S\) is a \(t\)-ideal in \(R_S\) for every multiplicatively closed subset \(S\) of \(R\) that is disjoint from \(I\). The authors proved that, every well-behaved \(t\)-ideal of \(R\) is contained in a maximal well-behaved \(t\)-ideal, and that \(R =\bigcap_{P\in \mathcal{P}} R_P\), where \(\mathcal{P}\) is the set of maximal well-behaved \(t\)-ideals. Also, they introduced a new star operation on \(R\), given by \(I^\blacklozenge=\bigcap_{P\in \mathcal{P}} IR_P\) for each nonzero fractional ideal \(I\) of \(R\). Using this new star operation, the authors proved that every maximal \(t\)-ideal of \(R\) is well-behaved if and only if \(\blacklozenge\leq t\), that is, \(I^\blacklozenge\subseteq I_t\) for each nonzero fractional ideal \(I\) of \(R\). Also, the authors proved a condition which is necessary for well-behavedness of \(R\) to pass to the polynomial ring \(R[X]\), and they gave an example to show that the condition need not hold. Finally, the authors showed that
1.
if \(R\) is locally a PvMD, then \(R\) is a PvMD if and only if the star operations \(\blacklozenge, w, t\) coincide on \(R\),
2.
for a height-one prime \(Q\) in an almost Krull domain, \(Q\) is divisorial if and only if \(Q\) contains a divisorial ideal contained in no other height-one prime if and only if \(Q\) is \(\blacklozenge\)-invertible.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13G05 Integral domains

Citations:

Zbl 0705.13001
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Full Text: DOI

References:

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