Mori domain if and only if every $v$-ideal is of finite type?

Question

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

Answer 1

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.