Abstract

The following statements for an infra-Krull domain <TEX>$R$</TEX> are shown to be equivalent: (1) <TEX>$R$</TEX> is a Krull domain; (2) for any essentially finite <TEX>$w$</TEX>-module <TEX>$M$</TEX> over <TEX>$R$</TEX>, the torsion submodule <TEX>$t(M)$</TEX> of <TEX>$M$</TEX> is a direct summand of <TEX>$M$</TEX>; (3) for any essentially finite <TEX>$w$</TEX>-module <TEX>$M$</TEX> over <TEX>$R$</TEX>, <TEX>$t(M){\cap}pM=pt(M)$</TEX>, for all maximal <TEX>$w$</TEX>-ideal <TEX>$p$</TEX> of <TEX>$R$</TEX>; (4) <TEX>$R$</TEX> satisfies the <TEX>$w$</TEX>-radical formula; (5) the <TEX>$R$</TEX>-module <TEX>$R{\oplus}R$</TEX> satisfies the <TEX>$w$</TEX>-radical formula.

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