In commutative alebra, I proved that in a Mori domain, every $v$-ideal(divisorial ideal) is of finite type. But converse is hard to me..I don't know it's correct. Someone help me plz
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You want to go slightly further and show that in a Mori domain, every $t$-ideal is of finite type. The argument that finite-type $t$-ideals imply Mori is simple, and essentially the same as the argument in, say, Noetherian iff ACC on all ideals. The important observation here is that
a union of a chain of divisorial ideals is always a $t$-ideal, in any domain.
Note that finite type $t$-ideals is enough to force the $v$-operation and $t$-operation to coincide.
In general, finite type $v$-ideals is not nearly enough to force a domain to Mori.
Badam Baplan
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Thanks for your great answer! Is $R=\mathbb{Z}+X\mathbb{Q}[[X]]$ an counterexample?. I know $R$ is not a Mori, but it's difficult to find $v$-ideal of $R$ – Silement May 27 '19 at 09:42