A Tychonoff space X such that the maximum ring of quotients of C(X) is uniformly complete is called a uniform quotients space. It is shown that this condition is equivalent to the Dedekind– MacNeille completion of C(X) being a ring of quotients of C(X), in the sense of Utumi. A compact metric space is a uniform quotients space precisely when it has a dense set of isolated points. Extremally dis...

In 1977 Hartwig and Luh asked whether an element a in a Dedekind-finite ring R satisfying aR = a2R also satisfies Ra = Ra2. In this paper, we answer this question in the negative. We also prove that if a is an element of a Dedekind-finite exchange ring R and aR = a2R, then Ra = Ra2. This gives an easier proof of Dischinger’s theorem that left strongly π-regular rings are right strongly π-regula...

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res(P,Pi) of the minimal polynomial P of a primitive integral element and of its irreducible factors Pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the ca...

It is proved that a ring A right or left Noetherian, distributive, centrally essential if and only [Formula: see text], where each of the rings text] either commutative Dedekind domain Artinian, uniserial ring.

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

‡ Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥min(n,N), we determine the number of functions (as...