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Elasticity of \(A+XB[X]\) when \(A\subset B\) is a minimal extension of integral domains. (English) Zbl 1228.13002

Let \(D\) be an atomic integral domain, that is, every nonzero nonunit of \(D\) is a finite product of irreducible elements or atoms. Then the elasticity of D is \(\rho (D)=\sup \{m/n|x_{1}\cdots x_{n}=y_{1}\cdots y_{m}\), \(x_{i},y_{j}\text{ irreducible }\}\). Here \(1\leq \rho (D)\leq \infty \) with \(\rho (D)=1\) if and only if any two atomic factorizations of a given element have the same length, that is, \(D\) is half-factorial. Let \((A,m)\) be a quasilocal integral domain with maximal ideal \(m\) that is not a field, let \(A\subset B\) be a minimal extension of integral domains, and let \(R=A+XB[X]\). The purpose of this paper is to compute \(\rho (R)\). Now either \(A\) is integrally closed in \(B\) or the extension is integral. If \(A\)is integrally closed in \(B\), then \(R\) is never atomic; so we assume that the extension \(A\subset B\) is integral. Then if \(B\) satisfies ACCP and there are two maximal ideals of \(B\) lying over \(M\), then \(\rho (R)=\infty \). Now suppose that \((B,M)\) is quasilocal. If \(m=M\) and \(B[X]\) is atomic, then \(\rho (R)=\rho (B[X])\). If \(M\neq m\) and \(B\) satisfies ACCP, then \(3/2\leq \rho (R)\leq 3\rho (B[X])\) and if \(B\)is a UFD we actually have \(\rho (R)=3/2\).

MSC:

13A05 Divisibility and factorizations in commutative rings
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13F20 Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

elasticity
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