Arithmetical rings and Krull dimension

Let A be a commutative arithmetical ring. It is proved that the ring A has Krull dimension if and only if every factor ring of A is finite-dimensional and does not have idempotent proper essential ideals.

Some Properties of Completely Arithmetical Rings

In this paper, we introduce the concept of completely arithmetical rings and investigate their properties. In particular, we prove that if R is a completely arithmetical ring with J(R)=0, then K0(R) ≅ ℤn for some positive integer n. We also show that such a ring is precisely a ring in which every proper ideal can be written uniquely as a product of finitely many distinct completely strongly irreducible ideals.