The Generalized Principal Ideal Theorem is one of the cornerstones of dimension theory for Noetherian rings. For an R-module M, we identify certain submodules of M that play a role analogous to that of prime ideals in the ring R. Using this definition, we extend the Generalized Principal Ideal Theorem to modules.

In this article we introduce the concept of $z^circ$-filter on a topological space $X$. We study and investigate the behavior of $z^circ$-filters and compare them  with corresponding ideals, namely, $z^circ$-ideals of $C(X)$,  the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^circ$-filter ...

The main theorem says that any module-finite (but not necessarily commutative) algebra Λ over a commutative Noetherian universally catenary ring R is catenary. Hence the ring Λ is catenary if R is Cohen-Macaulay. When R is local and Λ is a Cohen-Macaulay R-module, we have that Λ is a catenary ring, dim Λ = dimΛ/Q + htΛQ for any Q ∈ Spec Λ, and the equality n = htΛQ− htΛP holds true for any pair...

Introduction. There has arisen in recent years a substantial body of work on “multiplier ideals” in commutative rings (see [La]). Multiplier ideals are integrally closed ideals with properties that lend themselves to highly interesting applications. One is tempted then to ask just how special multiplier ideals are among integrally closed ideals in general. In this note we show that in a two-dim...

*Correspondence: [email protected] School of Mathematics Science, Liaocheng University, Liaocheng, Shandong 252059, China Abstract In order to generalize the notions of a (∈,∈ ∨q)-fuzzy subring and various (∈,∈ ∨q)-fuzzy ideals of a ring, a (λ,μ)-fuzzy subring and a (λ,μ)-fuzzy ideal of a ring are defined. The concepts of (λ,μ)-fuzzy semiprime, prime, semiprimary and primary ideals are introdu...

Let K be a number field with positive unit rank, and let OK denote the ring of integers of K. A generalization of Artin’s primitive root conjecture is that that OK is a primitive root set for infinitely many prime ideals. We prove this with additional conjugacy conditions in the case when K is Galois with unit rank greater than three. This was previously known under the assumption of the Genera...

Let U be the set of prime ideals P completion a Stanley–Reisner ring S, such that localization at Frobenius algebra injective hull residue field S is finitely generated algebra. We give partial answer to conjecture made by M. Katzman about openness U. Specifically, we show has non-empty interior and present some sufficient conditions for principal open D(f) contained in U, intersections closed ...

Solution: The ring F [x] of polynomials with coefficients in a field F is a P.I.D. Each prime ideal is generated by a monic, irreducible polynomial. Assume there are only a finite number of prime ideals generated by the polynomials f1, . . . , fn and let f(x) = 1+f1(x) · · · fn(x). No fi divides f , hence f is also irreducible. This contradicts the assumption that all the prime ideals were gene...

The prime and primitive spectra of Oq(kn), the multiparameter quantized coordinate ring of affine n-space over an algebraically closed field k, are shown to be topological quotients of the corresponding classical spectra, specO(k) and maxO(k) ≈ k, provided the multiplicative group generated by the entries of q avoids −1.