Variations of essentiality of ideals in commutative rings

In this paper, we describe how intersections with a totality of some ideals affect the essentiality of an ideal. We mainly study intersections with every (a) annihilator ideal, (b) prime ideal (c) strongly irreducible ideal (d) irreducible ideal and every pure ideal. After some general results, the paper focuses on [Formula: see text] to characterize spaces [Formula: see text] when every irreducible ideal of [Formula: see text] is pseudoprime. We also characterize the rings of continuous functions [Formula: see text] in which every pseudoprime ideal is strongly irreducible. We give a negative answer to a question raised by Gilmer and McAdam.

Maximal closed ideals of the Colombeau Algebra of Generalized functions

In this paper we investigate the structure of the set of maximal ideals of $${{\mathcal {G}}}(\Omega )$$ G ( Ω ) . The method of investigation passes through the use of the $$m-$$ m - reduction and the ideas are analoguous to those in Gillman and Jerison (Rings of Continuous Functions, N.J. Van Nostrand, Princeton, 1960) for the investigation of maximal ideals of continuous functions on a Hausdorff space K.

A class of ideals in intermediate rings of continuous functions

<p>For any completely regular Hausdorff topological space X, an intermediate ring A(X) of continuous functions stands for any ring lying between C^∗(X) and C(X). It is a rather recently established fact that if A(X) ≠ C(X), then there exist non maximal prime ideals in A(X).We offer an alternative proof of it on using the notion of z◦-ideals. It is realized that a P-space X is discrete if and only if C(X) is identical to the ring of real valued measurable functions defined on the σ-algebra β(X) of all Borel sets in X. Interrelation between z-ideals, z◦-ideal and Ʒ_A-ideals in A(X) are examined. It is proved that within the family of almost P-spaces X, each Ʒ_A -ideal in A(X) is a z◦-ideal if and only if each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).</p>