The Dedekind-Mertens formula and determinantal rings

We give a combinatorial proof of the Dedekind-Mertens formula by computing the initial ideal of the content ideal of the product of two generic polynomials. As a side effect we obtain a complete classification of the rank Cohen-Macaulay modules over the determinantal rings .

     

Galois G-covering of quotients of linear categories

In this paper, we introduce the notion of G-liftable ideals, which extends the liftable ideas defined by Assem and Le Meur. We characterize the G-liftable ideals and construct the Galois G-coverings of quotients of categories associated to the G-liftable ideals. In particular, we study the behavior of G-liftable admissible ideals under Galois G-coverings. Furthermore, we show that the ideals generated by finite dimensional projective modules over a locally bounded linear categories are admissible G-liftable ideals. As an application, we provide a reduction technique for dealing with the existence of Serre functors in the stable categories of Gorenstein projective objects.     

On Rings Whose Reflexive Ideals are Principal

We call a prime Noetherian maximal order R a pseudo-principal ring if every reflexive ideal of R is principal. This class of rings is a broad class properly containing both prime Noetherian pri-(pli) rings and Noetherian unique factorization rings (UFRs). We show that the class of pseudo-principal rings is closed under formation of n × n full matrix rings. Moreover, we prove that if R is a pseudo-principal ring, then the polynomial ring R[x] is also a pseudo-principal ring. We provide examples to illustrate our results.     

在环面同态作用下零维理想的性质

本文以矩阵变换和整理论为工具，研究了在环面同态作用下零维理想的性质，证明了零维理想的环面同态矩阵是满秩的，并且刻画了零维理想在环面作用下的对应关系．     

环的B-稳定正则理想

研究了正则理想是B-稳定的充分和必要条件,并且证明环R的正则理想I是B-稳定的当且仅当对任意的有限生成投射右R-模A,如果A₁和A₂是A的有限生成子模且满足A₁≌A₂,A₁=A₁I以及A₂=A₂I,则存在一个有限生成子模B,使得A=A₁（?）B=A₂（?）B;当且仅当对任意的幂等元e,f∈I,eR≌fR蕴含eR/（eR∩fR）≌fR/（eR∩fR）;当且仅当对任意的a∈1+I,存在一个幂等元e∈I,使得a-e∈∪（R）并且aR∩eR=0.进而构造了相关的例子.     

Algebras between C *(X) and C(X) that are closed under countable composition

Let be the algebra of all real-valued continuous functions on a completely regular space X, and the subalgebra of bounded functions. We show that the space of real maximal ideals of an intermediate algebra between and is a -embedded closed subspace of the product of with a product of copies of the real line. We make use of this embedding to provide a new characterization of the intermediate algebras that are closed under countable composition, and to exhibit an example of an intermediate algebra on that is closed under countable composition but not isomorphic to any .     

On Rings for Which Finitely Generated Ideals Have Only Finitely Many Minimal Components

For a commutative ring R we investigate the property that the sets of minimal primes of finitely generated ideals of R are always finite. We prove this property passes to polynomial ring extensions (in an arbitrary number of variables) over R as well as to R-algebras which are finitely presented as R-modules.     

Congruences and Ideals of Effect Algebras

This paper is devoted to a general investigation of congruences and ideals in effect algebras. One of our main results is the existence of an order isomorphism between Riesz congruences and Riesz ideals. We also answer an open question of Dvureenskij and Pulmannová by showing that an ideal is a Riesz ideal if and only if it is closed under generalized Sasaki projections.     

在模情况时二面体群$D_{2p}$作用下的transfer理想

In this paper, we determine the structures of the transfer ideal and its radical ideal for the ring of polynomials F p[x, y] under the action of dihedral group D2 p in the modular case. We mainly use Transfer variety, p order elements, and Hilbert’s Nullstellensatz Theorem.     

There do not exist minimal algebras between C*(X) and C(X) with prescribed real maximal ideal spaces

Let C(X) be the algebra of all real-valued continuous functions on a completely regular Hausdorff space X, and C*(X) the subalgebra of bounded functions. We prove that for any intermediate algebra A between C*(X) and C(X), other than C*(X), there exists a smaller intermediate algebra with the same real maximal ideals as in A. The space X is called A-compact if any real maximal ideal in A corresponds to a point in X. It follows that, for a noncompact space X, there does not exist any minimal intermediate algebra A for which A is A-compact. This completes the answer to a question raised by Redlin and Watson in 1987.     

Full Ideals

Contractedness of 𝔪-primary integrally closed ideals played a central role in the development of Zariski's theory of integrally closed ideals in two-dimensional regular local rings (R, 𝔪). In such rings, the contracted 𝔪-primary ideals are known to be characterized by the property that I: 𝔪 = I: x for some x ∈ 𝔪 ?𝔪². We call the ideals with this property full ideals and compare this class of ideals with the classes of 𝔪-full ideals, basically full ideals, and contracted ideals in higher dimensional regular local rings. The 𝔪-full ideals are easily seen to be full. In this article, we find a sufficient condition for a full ideal to be 𝔪-full. We also show the equivalence of the properties full, 𝔪-full, contracted, integrally closed, and normal, for the class of parameter ideals. We then find a sufficient condition for a basically full parameter ideal to be full.     

t-Schreier Domains

We study, under the name t-Schreier, the class of those integral domains whose group of t-invertible t-ideals satisfies the Riesz interpolation property. The so-called Prüfer v-multiplication domains (PVMDs) and Prüfer domains are special cases of t-Schreier domains. We show that, while a number of results known for Prüfer domains and PVMDs hold for these domains, the t-Schreier domains have a remarkable capability of unifying various results in that the results proved for t-Schreier domains can also be translated to results on pre-Schreier domains and hence on GCD and Bezout domains.     

Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

Nilpotent right alternative rings and bol circle loops

We determine the nilpotent right alternative rings of prime power oirder pⁿ n ≥ 4, which are not left alternative. Those which are strongly right alternative become Bol loops under the circle operation. The smallest Bol circle loop has order 16. There are six such loops, all of which appear to be new.     

Banach spaces of operators that are complemented in their biduals

Let [A, a] be a normed operator ideal. We say that [A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T _i ) _i∈I in A(X, Y) satisfying lim _i 〈y*, T _i x y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when [A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.      

The decision of prime and primary ideal

We give more efficient criteria to characterise prime ideal or primary ideal. Further, we obtain the necessary and sufficient conditions that an ideal is prime or primary in real field from the Gröbner bases directly.     

理想簇和根理想

关于环的理想的根有两种定义,一种是所有包含I的极大理想的交,另一种是所有包含I的素理想交,本文主要研究后者定义的一些性质,以及和理想簇V(I)(所有包含I的素理想的集合)的关系.     

Geometric interpretation of tight closure and test ideals

We study tight closure and test ideals in rings of characteristic using resolution of singularities. The notions of -rational and -regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal -Gorenstein ring of characteristic , the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings.

     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.