Let R be a right and left Ore ring, S its set of regular elements and Q = R[S −1 ] = [S −1 ]R the classical ring of quotients of R. We prove that if F. dim(Q Q ) = 0, then the following conditions are equivalent: (i) Flat right R-modules are strongly flat. (ii) Matlis-cotorsion right R-modules are Enochscotorsion. (iii) h-divisible right R-modules are weak-injective. (iv) Homomorphic images of weak-injective right R-modules are weak-injective. (v) Homomorphic images of injective right R-modules are weak-injective. (vi) Right R-modules of weak dimension ≤ 1 are of projective dimension ≤ 1. (vii) The cotorsion pairs (P 1 , D) and (F 1 , WI) coincide. (viii) Divisible right R-modules are weak-injective. This extends a result by Fuchs and Salce (2017) for modules over a commutative ring R.

No abstract

No abstract

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

No abstract