LCM-stableness in ring extensions
\textit{R. Gilmer} [in Ring Theory, Proc. Oklahoma Conf. 1973, Lect. Notes Pure Appl. Math. 7, 47-61 (1974; Zbl 0275.13019)] introduced the concept of LCM-stableness relating to GCD-properties of a commutative group ring. The main purpose of this paper is to point out that, in some cases, the necessary and sufficient conditions for a ring extension to be LCM-stable can be given in terms of polynomial grade, originally due to M. Hochster and developed by D. G. Northcott. Various different situations are considered and we shall attempt to give only a flavour of the results obtained. To be more explicit we fix the following notation. Let \(A\subset B\) be integral domains, K the quotient field of A and \(\Omega\) the algebraic closure of K. If I is an ideal of A let Gr(I) denote the polynomial grade of I. Let X denote an indeterminate and for any ideal J of the polynomial ring A[X], let c(J) denote the ideal of A generated by all coefficients of all polynomials in J. Finally if \(\alpha\in \Omega\) let \(A_{\alpha}\) denote the kernel of the canonical homomorphism of A[X] onto A[\(\alpha]\). Then \(A\subset B\) is called LCM-stable provided \((aA\cap bA)B=aB\cap bB\) for all a,b in A. In particular, if \(A\subset B\) is flat then \(A\subset B\) is LCM-stable, and the converse is true if B is a subring of K. Moreover, it is easily proved that \(A\subset B\) is LCM- stable provided \(A\subset A[x,y]\) is LCM-stable for all x,y in B. Now let 0\(\neq \alpha \in \Omega\) with \(\alpha^ m=u\in K\) for some positive integer m. Put \(u=a/b\) where 0\(\neq a,b\in A\) and assume that \(bX^ m-a\) is irreducible over K and \(A_ u\) has a linear base. Then \(A\subset A[\alpha]\quad is\quad LCM-stable\Leftrightarrow A\subset A[\alpha]\quad is\quad INC\Leftrightarrow A\subset A[\alpha]\quad is\quad flat\Leftrightarrow(a,b)\quad is\quad invertible.\) The author introduces two further ideas namely \(G_ 2\)-stableness and \(R_ 2\)-stableness and shows how these are related to LCM-stableness. He says that \(A\subset B\) is \(G_ 2\)-stable if Gr(IB)\(\geq 2\) for each non-zero finitely generated ideal I of A with Gr(I)\(\geq 2\). On the other hand \(A\subset B\) is \(R_ 2\)-stable if \(a:_ Bb=a\) for any 0\(\neq a,b\in A\) with \(a:_ Ab=a\). If \(A\subset B\) is LCM-stable then \(A\subset B\) is \(R_ 2\)-stable and the converse is true if A is a GCD-domain. Moreover if \(A\subset B\) is \(G_ 2\)-stable then \(A\subset B\) is \(R_ 2\)-stable and the converse is true if A is a Noetherian domain or a GCD-domain. It is proved that \(A\subset B\quad is\quad G_ 2-stable\Leftrightarrow A[X]\subset B[X]\quad is\quad G_ 2-stable\Leftrightarrow A[X]\subset B[X]\quad is\quad R_ 2- stable.\) In addition, if A is a GCD-domain then \(A\subset B\) is LCM- stable if and only if A[X]\(\subset B[X]\) is LCM-stable. Next let f(X) denote a prime element of A[X] of degree at least 1 and \(A_ f\) the ring A[X]/f(X)). Then \(A\subset A_ f\) is LCM-stable if and only if Gr(c(f))\(\geq 3\), provided A is a GCD-domain. One consequence of this fact is that if A is locally a GCD-domain and \(\alpha\) a non-zero element of \(\Omega\) then \(A\subset A[\alpha]\) is LCM-stable if and only if \(Gr(c(A_{\alpha}))\geq 3\). The paper contains quite a long discussion of when \(A\subset A[\alpha,\beta]\) is LCM-stable where \(\alpha\),\(\beta\) are non-zero elements of \(\Omega\). For example, in the case when A is locally a GCD-domain, \(A\subset A[\alpha]\) and \(A\subset A[\beta]\) are both LCM- stable and \(K(\alpha)\), \(K(\beta]\) are linearly disjoint over K then \(A\subset A[\alpha,\beta]\) is LCM-stable if and only if \(Gr(c(A_{\alpha})+c(A_{\beta}))\geq 4\). Numerous examples are given throughout the paper to complement the results proved.
LCM-stable extension of integral domains
scientific article; zbMATH DE number 3842017