## Completely integrally closed domains and t-ideals.(English)Zbl 0677.13007

For a domain A, let D(A), $$D_ f(A)$$, t(A), T(A) be the sets of fractional ideals which are respectively divisorial, v-finite divisorial, t-ideals, t-invertible t-ideals. Let $$D_ m(A), t_ m(A)$$ be the sets of ideals maximal among the proper integral ideals in D(A), t(A). Characterisations of a prime P being in T(A) include $$(1)\quad P\in D_ m(A)\cap D_ f(A)$$ and P:P$$\neq A:P$$, $$(2)\quad P\in t_ m(A)\cap D(A)$$ and P is v-invertible.
When, as in many of the results, A is c.i.c. (completely integrally closed) a prime P is in D(A) if and only if A:P$$\neq A$$; also, $$P\in T(A)\Leftrightarrow P\in D_ f(A)\Leftrightarrow P\in t_ m(A)\cap D(S).$$ From the many other results we cite that $$A\quad is\quad Krull\quad \Leftrightarrow \quad A\quad is\quad c.i.c.\quad and\quad t_ m(A)\subseteq D(A).$$ Related work includes (e.g.) V. Barucci and the author [J. Algebra 108, 161-173 (1987; Zbl 0623.13009)].
Reviewer: C.P.L.Rhodes

### MSC:

 13G05 Integral domains 13A15 Ideals and multiplicative ideal theory in commutative rings

Zbl 0623.13009