On \(t\)-reductions of ideals in pullbacks
Reductions considered in the article are generalizations of the concept introduced by \textit{D. G. Northcott} and \textit{D. Rees} [Proc. Camb. Philos. Soc. 50, 145--158 (1954; Zbl 0057.02601)]. They are related but distinct from \textit{N. Epstein}'s \(c\)-reductions [J. Algebra 323, No. 8, 2209--2225 (2010; Zbl 1243.13002)].NEWLINENEWLINELet \(R\) be a domain and \(I\) a nonzero fractional ideal of \(R\), then the \(v-\), \(t-\), and \(w-\) closures are defined respectively as \(I_v=\left(I^{-1}\right)^{-1}\), \(I_t=\bigcup J_{v}\), where \(J_v\) ranges over the set of finitely generated subideals of \(I\), and \(I_w=\bigcap IR_M\), where \(M\) ranges over the set of maximal \(t-\)ideals of \(R\). An ideal \(J\) is a \(*-\)reduction of \(I\) if \(\left(JI^n\right)^{*}=\left(I^{n+1}\right)^{*}\). An ideal that has no \(*-\)reductions but itself is called \(*-\)basic and a ring has \(*-\)basic ideal property if all its finitely generated ideals are \(*-\)basic, with \(v\), \(t\) or \(w\) substituted for \(*\). The notions of reduction and \(t\)-reduction coincide for Prüfer domains and the authors generalize this result by find necessary and sufficient conditions for the reduction and \(t\)-reduction to coincide in pseudo-valuation domains. In addition the article contains an example when the reduction and \(t\)-reduction of an ideal are distinct.NEWLINENEWLINEThe main result of the article states that if \(R\) is a domain then the notions of \(t\)-basic and \(v\)-basic ideals coincide, which is a generalization of results by \textit{J. H. Hays} [Trans. Am. Math. Soc. 177, 51--63 (1973; Zbl 0266.13001); Proc. Am. Math. Soc. 52, 81--84 (1975; Zbl 0345.13013)], and solves an open problem stated in [\textit{E. Houston} et al., ``\(*\)-reductions of ideals and Prüfer \(v\)-multiplication domains'', Preprint, \url{arXiv:1602.07035}].NEWLINENEWLINEFollowing the pullback diagram admitted by pseudo-valuation domains, the authors consider the following diagram denoted as \(\square\): NEWLINE\[NEWLINE\begin{aligned} R=\varphi^{-1}(D)\rightarrow D\\ \downarrow \downarrow\\ T\mathop\varphi^{\rightarrow} K:=T/M \end{aligned}NEWLINE\]NEWLINE where \(T\) is a domain, \(M\) is a maximal ideal of \(T\) with residue field \(K\), \(D\) is a proper subring of \(D\) with residue field \(k\). The article contains necessary and sufficient conditions for a pullback of type \(\square\) issued from local domains to inherit the finite \(t\)-basic (respectively \(v\)-basic) ideal property. In addition, the article furnished examples where those notions are not equivalent.
scientific article; zbMATH DE number 6636848
scientific article