Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

Coefficient Ideal of Ideals Generated by Monomials

In a commutative Noetherian ring R, the coefficient ideal of I relative to J is the largest ideal 𝔟 for which I𝔟 =J𝔟 when I is integral over J. In this article, we will give a simple algorithm to compute 𝔞(I, J) when I, J are ideals in a polynomial ring R = k[X ₁,…, X _d ] generated by monomials and J is a parameter ideal. We use the concept of socle sequence. Also we will show that the reduction number r _J (I) is also computed by our algorithm.     

Injective modules and torsion functors*

A commutative ring is said to have ITI with respect to an ideal 𝔞 if the 𝔞-torsion functor preserves injectivity of modules. Classes of rings with ITI or without ITI with respect to certain sets of ideals are identified. Behavior of ITI under formation of rings of fractions, tensor products, and idealization is studied. Applications to local cohomology over non-noetherian rings are given.     

The Ideal Completion of a Noetherian Local Domain

The ideal topology on a integral domain R is the linear topology which has as a fundamental system of neighborhoods of 0 the nonzero ideals of R. We investigate the properties of the ideal topology on a Noetherian local domain (R, 𝔪), and we establish connections between the 𝔪-adic completion and the ideal completion. We give conditions under which the completion in the ideal topology is Noetherian, and we show that, unlike the 𝔪-adic completion, the completion in the ideal topology is not always Noetherian.     

The Annihilating-Ideal Graph of a Commutative Ring with Respect to an Ideal

For a commutative ring R with identity, the annihilating-ideal graph of R, denoted 𝔸𝔾(R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted 𝔸𝔾 _I (R). We discuss when 𝔸𝔾 _I (R) is bipartite. We also give some results on the subgraphs and the parameters of 𝔸𝔾 _I (R).     

Adic finiteness: Bounding homology and applications

We prove the versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring R of prime characteristic is regular if and only if for some proper ideal 𝔟 the derived local cohomology complex RΓ_𝔟(R) has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.     

Conjugation-Invariant Subspaces and Lie Ideals in Non-Selfadjoint Operator Algebras

It is shown that a weakly closed subspace S of a nest algebraA is closed under conjugation by invertible elements in A, thatis, a^–1Sa=S if and only if S is a Lie ideal. A similarresult holds for not-necessarily-closed subspaces of algebrasof infinite multiplicity. An explicit characterisation of weaklyclosed Lie ideals in a nest algebra is given.     

On ad-nilpotent b-ideals for orthogonal Lie algebras

Krattenthaler, Orsina and Papi provided explicit formulas for the number of ad-nilpotent ideals with fixed class of nilpotence of a Borel subalgebra of a classical Lie algebra. Especially for types A and C they obtained refined results about these ideals with not only fixed class of nilpotence but also fixed dimension. In this paper, we shall follow their algorithm to determine the enumeration of ad-nilpotent b-ideals with fixed class of nilpotence and dimension for orthogonal Lie algebras, i.e., types B and D.     

Finiteness and Skolem Closure of Ideals for Non-Unibranched Domains

A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 _{k, a}  = {f ∈ Int(D) | f(a) ∈ 𝔪 ^k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 _a  = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated.     

On Flatness and Completion for Infinitely Generated Modules over Noetherian Rings

Let A be a noetherian commutative ring, and let 𝔞 be an ideal in A. We study questions of flatness and 𝔞-adic completeness for infinitely generated A-modules. This is done using the notions of decaying function and 𝔞-adically free A-module.     

Non-linear maps preserving ideals on a parabolic subalgebra of a simple algebra

Let P be an arbitrary parabolic subalgebra of a simple associative F-algebra. The ideals of P are determined completely; Each ideal of P is shown to be generated by one element; Every non-linear invertible map on P that preserves ideals is described in an explicit formula.     

Polynomial maps in two variables with smoothness on one line

Let F be a polynomial map in two variables over the complex field with nonzero constant Jacobian. Abhyankar proved that if F is smooth on generic lines then F is invertible. We sharpen Abhyankar’s result by proving that if F(𝕃) is smooth on one line 𝕃 then F is invertible. Criterion for smoothness of F(𝕃) is also discussed.     

Factorization into Prime and Invertible Ideals

Commutative rings for which every ideal can be written as afinite product of invertible and prime ideals are characterized.     

Derivations with Invertible or Nilpotent Values on a Multilinear Polynomial

Let R be a prime ring with no non-zero nil one-sided ideals, d a nonzero derivation on R, and f(X₁,...,X_t) a multilinear polynomial not central-valued on R. Suppose d(f(x₁,...,x_t)) is either invertible or nilpotent for all x₁,...,x_t in some non-zero ideal of R. Then it is proved that R is either a division ring or the ring of 2 × 2 matrices over a division ring. This theorem is a simultaneous generalization of a number of results proved earlier.1991 Mathematics Subject Classification: primary 16W25, secondary 16R50, 16N60, 16U80     

半素PI-环的本质(单边)理想

本文系统研究了半素ＰＩ－环本质（单边）理想及最大商环的有关性质,从ＰＩ－理论中著名的 Ｒｏｗｅｎ非平凡相交定理出发,证明了半素 ＰＩ－环的本质（单边）理想包合一个本质两边理想（定理７,推论８）,建立了半素ＰＩ－环本质（单边）理想与其中心本质理想的内在联系（定理 ９,推论 １０）,并以此为基础,证明了半素 ＰＩ－环的最大商环是 ＰＩ的（定理 １２）,且半素 ＰＩ－环的每一正则元在其最大商环中是可逆的（定理 １４）．     

Factoring operators over Hilbert-Schmidt maps and vector measures

We study the structure of Banach spaces X determined by the coincidence of nuclear maps on X with certain operator ideals involving absolutely summing maps and their relatives. With the emphasis mainly on Hilbert-space valued mappings, it is shown that the class of Hilbert—Schmidt spaces arises as a ‘solution set’ of the equation involving nuclear maps and the ideal of operators factoring through Hilbert—Schmidt maps. Among other results of this type, it is also shown that Hilbert spaces can be characterised by the equality of this latter ideal with the ideal of 2-summing maps. We shall also make use of this occasion to give an alternative proof of a famous theorem of Grothendieck using some well-known results from vector measure theory.     

Ad-nilpotent ideals of a parabolic subalgebra

We extend the results of Cellini and Papi [P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra, J. Algebra 225 (2000) 130–140; P. Cellini, P. Papi, Ad-nilpotent ideals of a Borel subalgebra II, J. Algebra 258 (2002) 112–121] on the characterizations of ad-nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical simple Lie algebra, we give formulas for the number of these ideals.