Constructible Ideals

Based on the study of simplicial complexes, one may naturally define the constructible monomial ideals. We connect the square-free constructible ideal with the Stanley–Reisner ideal of the Alexander dual associated to a constructible simplicial complex. We give some properties of constructible ideals, and we compute the Betti numbers. We prove that all monomial ideals with linear quotients are constructible ideals. We also show that all constructible ideals have a linear resolution.     

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.     

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.     

Minimal Resolutions and the Homology of Matching and Chessboard Complexes

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras. Our motivation is two-fold: We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Björner, Lovasz, Vreica and ivaljevi. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes. The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers.     

New Classes of Set-theoretic Complete Intersection Monomial Ideals

Let Δ be a simplicial complex and χ be an s-coloring of Δ. Biermann and Van Tuyl have introduced the simplicial complex Δ_χ. As a corollary of Theorems 5 and 7 in their 2013 article, we obtain that the Stanley–Reisner ring of Δ_χ over a field is Cohen–Macaulay. In this note, we generalize this corollary by proving that the Stanley–Reisner ideal of Δ_χ over a field is set-theoretic complete intersection. This also generalizes a result of Macchia.     

A Characterization of Cancellation Ideals

An ideal of a commutative ring with identity is called a cancellation ideal if whenever for ideals and of , then . We show that an ideal is a cancellation ideal if and only if is locally a regular principal ideal.

     

Fuzzy Ideals and Fuzzy Interior Ideals in Ordered Semirings

The notions of fuzzy ideals and fuzzy interior ideals in ordered semirings are introduced and some of their characterizations are obtained through regularity criterion and with some operations on them.     

On n-Absorbing Ideals of Commutative Rings

Let R be a commutative ring with 1 ≠ 0 and n a positive integer. In this article, we study two generalizations of a prime ideal. A proper ideal I of R is called an n-absorbing (resp., strongly n-absorbing) ideal if whenever x ₁…x _n+1 ∈ I for x ₁,…, x _n+1 ∈ R (resp., I ₁…I _n+1 ? I for ideals I ₁,…, I _n+1 of R), then there are n of the x _i 's (resp., n of the I _i 's) whose product is in I. We investigate n-absorbing and strongly n-absorbing ideals, and we conjecture that these two concepts are equivalent. In particular, we study the stability of n-absorbing ideals with respect to various ring-theoretic constructions and study n-absorbing ideals in several classes of commutative rings. For example, in a Noetherian ring every proper ideal is an n-absorbing ideal for some positive integer n, and in a Prüfer domain, an ideal is an n-absorbing ideal for some positive integer n if and only if it is a product of prime ideals.     

Fuzzy理想和Fuzzy幂零根

本文给出了Ｆ—准素（ＦＷ—准素）理想的有限交是Ｆ－准素理想（ＦＷ－准素理想）的几个充分条件。此外，证明了［５］中定义的一个交换环Ｒ的Ｆｕｚｚｙ幂零根是Ｒ的幂零根Ｋ的特征函数Ｘ＿Ｋ。     

L-Fuzzy理想的新刻画

借助于水平截集给出 L -fuzzy子环 ,L -fuzzy理想和 L -fuzzy素理想的若干新刻画     

Ideals of noncommutative DRℓ-monoids

In this paper, we introduce the concept of an ideal of a noncommutative dually residuated lattice ordered monoid and we show that congruence relations and certain ideals are in a one-to-one correspondence.     

关于单状态变量的局部分岔问题中的内蕴理想

本文给出单状态变量的局部分岔问题中一类具有有限余维数的理想中最大内蕴理想的计算公式并且得到关于内蕴理想的两个充分必要条件.     

Structure Theory for Grade Three Perfect Ideals Associated with Some Matrices

Kang, Cho, and Ko gave a structure theorem for some classes of perfect ideals of grade 3 which are linked to almost complete intersections by a regular sequence. We define a 5 × 9 matrix f determined by three matrices A, T, and Y and construct a class of perfect ideals 𝒦₄(f) of grade 3 defined by f. This contains a class of perfect ideals of grade 3 minimally generated by five elements which are not Gorenstein. We give a structure theorem for some classes of perfect ideals of grade 3 linked to 𝒦₄(f) by a regular sequence in 𝒦₄(f).     

Regular CW-Complexes and Poset Resolutions of Monomial Ideals

Abstract

We study the generation of a finite group by its conjugacy classes, while generalizing basic concepts from linear algebra: basis and dimension. Besides the well known Burnside Basis Theorem for finite p-groups, there is no direct extension of these concepts to other families of finite groups. We show that by considering generating sets consisting of conjugacy classes, there is a possibility for such a generalization.     

Minimal Free Resolutions of Complete Bipartite Graph Ideals

This paper gives an explicit construction for the minimal free resolution of a complete bipartite graph ideal. This yields a combinatorial formula for the Betti numbers and projective dimension of complete bipartite graph ideals.     

Spectra of Maximal 1-Sided Ideals and Primitive Ideals

R is any ring with identity. Let Spec _r (R) (resp. Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all maximal right ideals, all right primitive ideals) of R and let U _r (eR) = {P ? Spec _r (R)| e ? P}. Let  = ∪_{P?Prim r (R)} Spec _r ^P (R), where Spec _r ^P (R) = {Q ?Spec _r ^P (R)|P is the largest ideal contained in Q}. A ring is called right quasi-duo if every maximal right ideal is 2-sided. In this article, we study the properties of the weak Zariski topology on  and the relationships among various ring-theoretic properties and topological conditions on it. Then the following results are obtained for any ring R: (1) R is right quasi-duo ring if and only if  is a space with Zariski topology if and only if, for any Q ? , Q is irreducible as a right ideal in R. (2) For any clopen (i.e., closed and open) set U in ? = Max _r (R) ∪  Prim _r (R) (resp.  = Prim _r (R)) there is an element e in R with e ² ? e ? J(R) such that U = U _r (eR) ∩  ? (resp. U = U _r (eR) ∩  ), where J(R) is the Jacobson of R. (3) Max _r (R) ∪  Prim _r (R) is connected if and only if Max _l (R) ∪  Prim _l (R) is connected if and only if Prim _r (R) is connected.     

BCH-代数的拟结合理想和拟结合模糊理想

在BCH-代数中,引入了拟结合理想和模糊拟结合理想的概念.研究了BCH-代数中模糊拟结合理想的相关性质,给出了BCH-代数拟结合理想的构造.     

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.