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


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


Zbl 0623.13009