ABSTRACT
We study commutative rings with the acc on irreducible ideals, denoted acci, and a weaker condition accsi, or the acc on subdirectly irreducible (= SDI) ideals. In an acci ring R any irreducible ideal is primary (Theorem 4.7). Moreover, any acci ring R has acci classical quotient Q(R), and a 1–1 correspondence between prime ideals and indecomposable injectives. We relate these conditions to various rings: SISI (see below), O-dimensional, Arithmetic, FP-injective, IF (= injectives are flat) and polynomial rings over them.
In particular, any locally Noetherian ring R, e.g. any commutative von Neumann regular (= VNR) ring R, and moreover any polynomial ring R[X] over R, is a accsi ring (Corollary 2.2). Furthermore, if Rp is Noetherian for all prime ideals R, then Q(R/I) is quasi-Frobenius and I is primary for any irreducible ideal I (Theorem 4.12). A similar conclusion holds for an Arithmetic acci ring R (Theorems 3.9 and 3.9′.)
Using a theorem of Facchini [F], in §5 we show that any acci Prüfer domain R is strongly discrete in the sense of Popescu [P]. (See Corollary 5.6). In particular, every irreducible ideal is then a power of its radical (op. cit.).
Any quotient finite dimensional acci ring satisfies acc on radical ideals (= Noetherian spec.) (Theorem 5.1). The proof uses the characterizations of the latter due to Ohm and Pendleton [O-P] and Pusat-Yilmaz and Smith [P-S].
Any uniform (= 0 is an irreducible ideal) acci ring R satisfies the acc on point annihilators (= accpa), and any irreducible accpa ring is primary (Theorem 4.7). A theorem of Cedó states that any accpa ring R has semilocal Q (R). Then, if R has finite Goldie dimension, or if | Ass R| is finite, then Q (R) is Kasch (Theorem 6.4).
