On prime ideals and radicals of polynomial rings and graded rings

We extend some known results on radicals and prime ideals from polynomial rings and Laurent polynomial rings to Z$\mathbb{Z}$-graded rings, i.e, rings graded by the additive group of integers. The main of them concerns the Brown–McCoy radical G$G$ and the radical S$S$, which for a given ring A$A$ is defined as the intersection of prime ideals I$I$ of A$A$ such that A/I$A/I$ is a ring with a large center. The studies are related to some open problems on the radicals G$G$ and S$S$ of polynomial rings and situated in the context of Koethe’s problem.     

Prime ideals of graded rings and related matters

Let R be a ring graded by an abelian group.We study prime ideals of R that are maximal for not containing nonzero homogeneous elements.Also prime ideals of the symmetric graded Martindale ring of quotients of R are investigated.The results are applied to study when R is a Jacobson ring in case R is a Z-graded ring or a group ring of a finitely generated abelian group, or in case R is right Noetherian and strongly graded by a polycyclic-by-finite group.     

The -invariant and Gorensteinness of graded rings associated to filtrations of ideals in regular local rings

Let be a regular local ring and let be a filtration of ideals in such that is a Noetherian ring with . Let and let be the -invariant of . Then the theorem says that is a principal ideal and for all if and only if is a Gorenstein ring and . Hence , if is a Gorenstein ring, but the ideal is not principal.

     

Rees short exact sequences of S-systems

In this article, the conditions for a Rees short exact sequence of S -systems to be left and right split, respectively, are given. We note that this result differs from the well known result of an exact sequence of modules that is split.     

Hilbert coefficients and the associated graded rings

Let be a -dimensional Cohen-Macaulay local ring with infinite residue field. Let be an -primary ideal of . In this paper, we prove that if for some minimal reduction of , then depth .

     

Equalities of ideals associated with two projections in rings with involution

In this article we study various right ideals associated with two projections (self-adjoint idempotents) in a ring with involution. Results of O.M. Baksalary, G. Trenkler, R. Piziak, P.L. Odell and R. Hahn about orthogonal projectors (complex matrices which are Hermitian and idempotent) are considered in the setting of rings with involution. New proofs based on algebraic arguments, rather than finite-dimensional and rank theory, are given.     

Phantom depth and flat base change

We prove that if is a flat local homomorphism, is Cohen-Macaulay and -injective, and and share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.

     

Hilbert coefficients and Buchsbaumness of associated graded rings

Let A be a Noetherian local ring with the maximal ideal and I an -primary ideal. The purpose of this paper is to generalize Northcott's inequality on Hilbert coefficients of I given in Northcott (J. London Math. Soc. 35 (1960) 209), without assuming that A is a Cohen-Macaulay ring. We will investigate when our inequality turns into an equality. It is related to the Buchsbaumness of the associated graded ring of I.     

Annihilator ideals,associated primes and kasch-mccoy commutative rings

If R is a ring, and A right annulet(= annihilator right ideal) then A[X] is a right annulet of the polynomial ring R[X]. (In factX can be any set of variables.). An annulet I of R[X] of this form is said to be be extended. Not all annulets of R[X] are extended, since e.g., the ascending chain on right annulets (= acc ┴) is not inherited by R[X], as, Kerr [Ke] observed. Nevertheless, maximal (minimal) annulets of a polynomial ring R[X] are extended, as a theorem of McCopy on annihilators in R[X] readily shows (see Introduction).     

Test ideals and base change problems in tight closure theory

Test ideals are an important concept in tight closure theory and their behavior via flat base change can be very difficult to understand. Our paper presents results regarding this behavior under flat maps with reasonably nice (but far from smooth) fibers. This involves analyzing, in depth, a special type of ideal of test elements, called the CS test ideal. Besides providing new results, the paper also contains extensions of a theorem by G. Lyubeznik and K. E. Smith on the completely stable test ideal and of theorems by F. Enescu and, independently, M. Hashimoto on the behavior of -rationality under flat base change.

     

One-dimensional local rings with reduced associated graded ring and their Hilbert functions

Let A be a one-dimensional reduced local ring with finite normalization. Let G(A) be the associated graded ring of A. In this paper we analyse the two conditions: Proj (G(A)) reduced and G(A) reduced together with their relations with the equality H(n)=H_R (n), where H(n) and H_R (n) are respectively the Hilbert function of the ring A and of the local ring R of G(A)_red=G(A)/nil (G(A)) at its homogenous maximal ideal. As a consequence of our results we get a class of ordinary singularities with H(n) locally decreasing for any embedding dimension H(1) greater then 4.