In this paper, we study a generalization of z-ideals in the ring C(X) of continuous real valued functions on a completely regular Hausdorff space X. The notion of a weak ideal and naturally a weak z-ideal and a prime weak ideal are introduced and it turns out that they behave such as z-ideals in C(X).

We introduced and studied GF-regular modules as a generalization of π-regular rings to modules as well as regular modules (in the sense of Fieldhouse). An R-moduleM is called GF-regular if for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rntrnx = rx. The notion of G-pure submodules was introduced to generalize pure submodules and proved that an RmoduleM isGF-regula...

This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued f...

We provide necessary and sufficient conditions for a Gaussian ring R to be semihereditary, or more generally, of w.dimR ≤ 1. Investigating the weak global dimension of a Gaussian coherent ring R, we show that the only values w.dimR may take are 0, 1 and ∞; but that fP.dimR is always at most one. In particular, we conclude that a Gaussian coherent ring R is either Von Neumann regular, or semiher...

We show that a ring is a Krull ring if and only if every nonzero regular prime ideal contains a t-invertible prime ideal if and only if every proper regular principal ideal is quasi-equal to a product of prime ideals.

In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a 2-dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a 2-dimensional regular local ring to have the same value semigroup is obtained.

We introduce the notion of clean ideal, which is a natural generalization of clean rings. It is shown that every matrix ideal over a clean ideal of a ring is clean. Also we prove that every ideal having stable range one of a regular ring is clean. These generalize the corresponding results for clean rings. 1. Introduction. Let R be a unital ring. We say that R is a clean ring in case every elem...

The concepts of Boolean metric space and convex combination are used to characterize polynomial maps An −→ Am in a class of commutative Von Neumann regular rings including p-rings, that we have called CFG-rings. In those rings, the study of the category of algebraic varieties (i.e. sets of solutions to a finite number of polynomial equations with polynomial maps as morphisms) is equivalent to t...

Recently, it has been proposed that the natural connectivity can be used to efficiently characterise the robustness of complex networks. Natural connectivity quantifies the redundancy of alternative routes in a network by evaluating the weighted number of closed walks of all lengths and can be regarded as the average eigenvalue obtained from the graph spectrum. In this article, we explore the n...

It is known that a linear two-weight code C over a finite field Fq corresponds both to a multiset in a projective space over Fq that meets every hyperplane in either a or b points for some integers a < b , and to a strongly regular graph whose vertices may be identified with the codewords of C . Here we extend this classical result to the case of a ring-linear code with exactly two nonzero homo...