Valuation domains with a maximal immediate extension of finite rank

Francois Couchot (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

If $R$ is a valuation domain of maximal ideal $P$ with a maximal immediate extension of finite rank it is proven that there exists a finite sequence of prime ideals $P=L_0\supset L_1\supset\dots\supset L_m\supseteq 0$ such that $R_{L_j}/L_{j+1}$ is almost maximal for each $j$, $0\leq j\leq m-1$ and $R_{L_m}$ is maximal if $L_m\ne 0$. Then we suppose that there is an integer $n\geq 1$ such that each torsion-free $R$-module of finite rank is a direct sum of modules of rank at most $n$. By adapting Lady's methods, it is shown that $n\leq 3$ if $R$ is almost maximal, and the converse holds if $R$ has a maximal immediate extension of rank $\leq 2$.

Mots clés: torsion-free module; valuation domain; strongly flat module; content module

Domaines: Anneaux et algèbres [math.RA]

Francois Couchot. Valuation domains with a maximal immediate extension of finite rank. Journal of Algebra, 2010, 323 (1), pp.32-41. ⟨hal-00352347⟩