In this article‎, ‎we have characterized ideals in $C(X)$ in which‎ ‎every ideal is also an ideal (a $z$-ideal) of $C(X)$‎. ‎Motivated by‎ ‎this characterization‎, ‎we observe that $C_infty(X)$ is a regular‎ ‎ring if and only if every open locally compact $sigma$-compact‎ ‎subset of $X$ is finite‎. ‎Concerning prime ideals‎, ‎it is shown that‎ ‎the sum of every two prime (semiprime) ideals of e...

Let R be a commutative integral domain with quotient field K and let P be a nonzero strongly prime ideal of R. We give several characterizations of such ideals. It is shown that (P : P) is a valuation domain with the unique maximal ideal P. We also study when P^{&minus1} is a ring. In fact, it is proved that P^{&minus1} = (P : P) if and only if P is not invertible. Furthermore, if P is invertib...

A natural generalization of Cohen's set of forcing conditions (the t w ~ valued functions with domain a finite subset of w) is the set of twovalued functions with domain an element of an ideal J on ~. The I~roblem treated in this paper is to determine when such forcing yields a generic real of minimal degree of constructibility. A :;imple decomposition argument shows that the non-maximality of ...

Introduction. L. Fuchs [2 ] has given for Noetherian rings a theory of the representation of an ideal as an intersection of primal ideals, the theory being in many ways analogous to the classical Noether theory. An ideal Q is primal if the elements not prime to Q form an ideal, necessarily prime, called the adjoint of Q. Primary ideals are necessarily primal, but not conversely. Analogous resul...

Let $R$ be a commutative ring with identity. Let $G(R)$ denote the maximal graph associated to $R$, i.e., $G(R)$ is a graph with vertices as the elements of $R$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $R$ containing both. Let $Gamma(R)$ denote the restriction of $G(R)$ to non-unit elements of $R$. In this paper we study the various graphi...

Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J . Let F = {In}n≥0 be a good filtration of ideals in A. Denote by FJ (F) = ⊕ n≥0 (In/JIn)t n the fiber cone of F with respect to J. The paper characterizes the multiplicity and the CohenMacaulayness of FJ (F) in terms of minimal reductions of F .

We extend the Gröbner basis theory developed in [10, 11] to certain non-homogeneous, locally filtered finitely generated ideals in R0 , and to certain admissible orders. The main tool used is the study of two homogeneous ideals that may be associated to an ideal I R0 , namely the ideal grT (I) generated by all homogenous components of maximal degree of elements in I , and the “homogenized” idea...

We investigate monomial labellings on cell complexes, giving a minimal cellular resolution of the ideal generated by these monomials, and such that the associated quotient ring is Cohen-Macaulay. We introduce a notion of such a labelling being maximal. There is only a finite number of maximal labellings for each cell complex, and we classify these for trees, partly for subdivisions of polygons,...