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Proceedings of the American Mathematical Society

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Nonsplitting sequences of value groups

Author: Joe L. Mott
Journal: Proc. Amer. Math. Soc. 44 (1974), 39-42
MSC: Primary 13D99
MathSciNet review: 0332770
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Abstract: If $ K$ is the quotient field of an integral domain $ D$, then the value group $ {V_K}(D)$ of $ D$ in $ K$ is the group $ {K^ \ast }/U(D)$, partially ordered by $ {D^ \ast }/U(D)$, where $ U(D)$ denotes the group of units of $ D$. This note shows that if the sequence

$\displaystyle (1)\quad \{ 1\} \to G \to H \to J \to \{ 1\} $

is lexicographically exact and if $ H$ is lattice ordered, then there is a Bezout domain $ B$ and a prime ideal $ P$ of $ B$ such that $ {V_K}(B) = H,{V_K}({B_P}) = J$, and $ {V_k}(B/P) = G$, where $ k$ denotes the residue field of $ {B_P}$. Moreover, $ B$ is the direct sum of $ B/P$ and $ P$, and $ {B_P} = k + P$. In particular, the sequence (1) need not split, even with somewhat stringent restrictions on the integral domain $ B$. This gives a negative answer to a question posed by R. Gilmer.

References [Enhancements On Off] (What's this?)

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Keywords: Ordered group, valuation, group algebra, lexicographically exact sequence of ordered groups, Bezout domain
Article copyright: © Copyright 1974 American Mathematical Society