On weak subintegrality (Q5961641)

Given an extension \(A\subseteq B\) of arbitrary commutative rings, there is a largest extension \(^+_BA\) of \(A\) in \(B\) such that \(A\subseteq {^+_BA}\) is subintegral, and a largest extension \(^*_BA\) such that \(A\subseteq {^*_BA}\) is weakly subintegral. The extensions \(^+_BA\) and \(^*_BA\) are respetively called the seminormalization and weak normalization of \(A\) in \(B\). An extension \(A\subseteq A[b]\) of commutative rings is called elementary subintegral if \(b^2\), \(b^3\in A\). The extension \(A\subseteq A[b]\) is called elementary weakly subintegral if \(b^q\), \(qb\in A\) for some positive prime \(q\).NEWLINENEWLINENEWLINEDefinition. Let \(A\subseteq B\) be an extension of commutative rings. Then: (1.1.1) \(b\in B\) is subintegral over \(A\) if \(A \subseteq A[b]\) is subintegral. (1.1.2) \(b\in B\) is weakly subintegral over \(A\) if \(A\subseteq A[b]\) is weakly subintegral.NEWLINENEWLINENEWLINE\textit{L. G. Roberts} and \textit{B. Singh} [Compos. Math. 85, No. 3, 249-279 (1993; Zbl 0782.13006)] proved that if \(\mathbb{Q} \subseteq A\) then (1.1.1) is equivalent to the following condition:NEWLINENEWLINENEWLINECondition 1.2.1. There exists \(p\in \mathbb{Z}^+\), elements \(c_1, \dots, c_p\in B\), and \(N\in\mathbb{N}\) such that \(b^n+ \sum^p_{i=1} {n\choose i} c_ib^{n-i} \in A\) for \(n\in \mathbb{N}\), \(n\geq N\).NEWLINENEWLINENEWLINESeveral modified versions of condition (1.2.1) appeared:NEWLINENEWLINENEWLINECondition 1.2.1f. There exist elements \(c_1, \dots, c_p\in B\) and \(N\in \mathbb{N}\) such that NEWLINE\[NEWLINEb^n+ \sum^p_{i=1} {n\choose i} c_ib^{n-i} \in A\text{ for }N\leq n\leq 2N+2 p-1.NEWLINE\]NEWLINE Condition 1.2.2 (1.2.1 with \(N=1)\). There exist elements \(c_1, \dots, c_p\in B\) such that NEWLINE\[NEWLINEb^n+ \sum^p_{i=1} {n\choose i} c_ib^{n-i} \in A\text{ for }n\geq 1.NEWLINE\]NEWLINE Condition 1.2.2f (1.2.1f with \(N=1)\). There exist elements \(c_1, \dots, c_p\in B\) such that NEWLINE\[NEWLINEb^n+ \sum^p_{i=1} {n\choose i} c_ib^{n-i} \in A\text{ for }1\leq n\leq 2p+1.NEWLINE\]NEWLINE Condition 1.2.3. There exist elements \(c_1, \dots, c_p\in B\) and \(N\in \mathbb{N}\), \(s\in \mathbb{Z}^+\), \(N\geq s+p\) such that NEWLINE\[NEWLINEb^n+ \sum^p_{i=1} {n\choose i} c_ib^{n-i-s} \in A\text{ for }n\geq N.NEWLINE\]NEWLINE Condition 1.3. There exist an integer \(p\geq 0\) and elements \(a_1, \dots, a_{2p+1} \in A\) such that NEWLINE\[NEWLINEb^n+ \sum^n_{i=1} (-1)^i {n\choose i} a_ib^{n-i} =0\text{ for }p+1\leq n\leq 2p+1.NEWLINE\]NEWLINE Main results: Theorem A. Conditions (1.2.*) are mutually equivalent and each of these is equivalent to condition 1.3.NEWLINENEWLINENEWLINETheorem B. Let \(A\subseteq B\) be an extension of arbitrary commutative rings and let \(b\in B\). Then the following are equivalent:NEWLINENEWLINENEWLINE(1) \(b\) is weakly subintegral over \(A\) (as defined in (1.1.2)). NEWLINENEWLINENEWLINE(2) \(b\) satisfies any of the (equivalent) conditions of theorem \(A\).NEWLINENEWLINENEWLINETheorem C. Let \(A\subset B\) be an extension of arbitrary commutative rings. Then \(^*_BA\) is the set of all elements of \(B\) satisfying the equivalent conditions of theorem B. In particular, \(A\subset B\) is weakly subintegral if and only if every element of \(B\) satisfies the conditions of theorem B.