Properties Of Ideals Research Articles

Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

Let $G$ be a graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro were the first to consider a monomial ideal $\mathcal{M}_G$, called the $G$-parking function ideal, in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively to $G$-parking functions. Dochtermann introduced and studied skeleton ideals of the graph $G$, which are subideals of the $G$-parking function ideal with an additional parameter $k ~(0\le k \le n-1)$. A $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of the graph $G$ is generated by monomials corresponding to non-empty subsets of the set of non-root vertices $[n]$ of size at most $k+1$. Dochtermann obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the $k$-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.

Maximal prime homomorphic images of mod-p Iwasawa algebras

AbstractLet k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–)α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.

Packing properties of cubic square-free monomial ideals

The symbolic powers, in general, are not equal to the ordinary powers. Therefore, one interesting question here is for what classes of ideals ordinary and symbolic powers coincide? The answer to this question for square-free monomial ideals may be packing property. In this paper, we classify all cubic path ideals for those the symbolic and ordinary powers coincide.

Cohomological dimension of ideals defining Veronese subrings

Given a standard graded polynomial ring over a commutative Noetherian ring A A , we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when A A is a field of characteristic zero, and follows from a result of Peskine and Szpiro when A A is a field of positive characteristic; our result applies, for example, when A A is the ring of integers.

Intuitionistic fuzzy normed prime and maximal ideals

<abstract><p>Motivated by the new notion of intuitionistic fuzzy normed ideal, we present and investigate some associated properties of intuitionistic fuzzy normed ideals. We describe the intrinsic product of any two intuitionistic fuzzy normed subsets and show that the intrinsic product of intuitionistic fuzzy normed ideals is a subset of the intersection of these ideals. We specify the notions of intuitionistic fuzzy normed prime ideal and intuitionistic fuzzy normed maximal ideal, we present the conditions under which a given intuitionistic fuzzy normed ideal is considered to be an intuitionistic fuzzy normed prime (maximal) ideal. In addition, the relation between the intuitionistic characteristic function and prime and maximal ideals is generalized. Finally, we characterize relevant properties of intuitionistic fuzzy normed prime ideals and intuitionistic fuzzy normed maximal ideals.</p></abstract>

Ideal theory on EQ-algebras

<abstract><p>In this article, we introduce ideals and other special ideals on EQ-algebras, such as implicative ideals, primary ideals, prime ideals and maximal ideals. At first, we give the notion of ideal and its related properties on EQ-algebras, and give its equivalent characterizations. We discuss the relations between ideals and filters, and study the generating formula of ideals on EQ-algebras. Moreover, we study the properties of implicative ideals, primary ideals, prime ideals and maximal ideals and their relations. For example, we prove that every maximal ideal is prime and if prime ideals are implicative, then they are maximal in the EQ-algebra with the condition $ (DNP) $. Finally, we introduce the topological properties of prime ideals. We get that the set of all prime ideals is a compact $ T_{0} $ topological space. Also, we transferred the spectrum of EQ-algebras to bounded distributive lattices and given the ideal reticulation of EQ-algebras.</p></abstract>

Cancellation ideals of a ring extension

We study properties of cancellation ideals of ring extensions. Let R⊆S be a ring extension. A nonzero S-regular ideal I of R is called a (quasi)-cancellation ideal of the ring extension R⊆S if whenever IB=IC for two S-regular (finitely generated) R-submodules B and C of S, then B=C. We show that a finitely generated ideal I is a cancellation ideal of the ring extension R⊆S if and only if I is S-invertible.

Approximation properties of tensor norms and operator ideals for Banach spaces

Abstract For a finitely generated tensor norm α \alpha , we investigate the α \alpha -approximation property ( α \alpha -AP) and the bounded α \alpha -approximation property (bounded α \alpha -AP) in terms of some approximation properties of operator ideals. We prove that a Banach space X has the λ \lambda -bounded α p , q {\alpha }_{p,q} -AP ( 1 ≤ p , q ≤ ∞ , 1 / p + 1 / q ≥ 1 ) (1\le p,q\le \infty ,1/p+1/q\ge 1) if it has the λ \lambda -bounded g p {g}_{p} -AP. As a consequence, it follows that if a Banach space X has the λ \lambda -bounded g p {g}_{p} -AP, then X has the λ \lambda -bounded w p {w}_{p} -AP.

On the dimension of ideals in group algebras, and group codes

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

Ideals and Bosbach States on Residuated Lattices

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

Some Properties of Q-Neutrosophic Ideals of Semirings

Some Properties of Q-Neutrosophic Ideals of Semirings

δ-Ideals in pseudo-complemented distributive join-semilattices

The concept of a [Formula: see text]-ideal is introduced in a pseudo-complemented distributive join semilattice with [Formula: see text] and some properties of these ideals are obtained. We also give a characterization for a pseudo-complemented distributive join-semilattice to be a Stone join-semilattice. Also, in a distributive join-semilattice a characterization for an ideal to be a [Formula: see text]-ideal is proved.

On ideals of rings of continuous functions associated with sublocales

Let X be a Tychonoff space. Associated with every subset A⊆βX is the ideal OA of the ring C(X) consisting of all functions that vanish in a neighborhood of X∩A. Now, viewing Tychonoff spaces as objects in the category CRLoc of completely regular locales, we have ideals of the form OA, where A is a sublocale of βX. In this paper we study properties of such ideals not only for Tychonoff spaces, but for any object in CRLoc. Carrying out the discussion in this category, we have more function rings (they are denoted RL, for any L∈CRLoc) than the class of the rings C(X). Pure ideals of C(X) are known to be exactly the ideals OA, for A a closed subset of βX. We characterize the spaces for which the ideals OA, for A a closed subset of X (note that it need not be closed in βX) are pure ideals. They properly contain the normal ones. We describe the socle of any ring RL as the ideal OA, with A equal to the join of all nowhere dense sublocales of βL. We show that the socle is zero precisely when βL is dense in itself, and essential if and only if the smallest dense sublocale of βL has a complement in the lattice of sublocales of βL. If βL is scattered, then this is so if and only if L has a smallest nowhere dense sublocale.

Regularity and ℎ-polynomials of toric ideals of graphs

For all integers 4 ≤ r ≤ d 4 \leq r \leq d , we show that there exists a finite simple graph G = G r , d G= G_{r,d} with toric ideal I G ⊂ R I_G \subset R such that R / I G R/I_G has (Castelnuovo–Mumford) regularity r r and h h -polynomial of degree d d . To achieve this goal, we identify a family of graphs such that the graded Betti numbers of the associated toric ideal agree with its initial ideal, and, furthermore, that this initial ideal has linear quotients. As a corollary, we can recover a result of Hibi, Higashitani, Kimura, and O’Keefe that compares the depth and dimension of toric ideals of graphs.

Betti numbers of edge ideals of some split graphs

In this article, we shall compute the graded Betti numbers of edge ideals of complete split and nearly complete split graphs. Also, we obtain the bounds on the Betti numbers and the projective dimension of edge ideals of split graphs.

Free resolutions of function classes via order complexes

Function classes are collections of Boolean functions on a finite set, which are fundamental objects of study in theoretical computer science. We study algebraic properties of ideals associated to function classes previously defined by the third author. We consider the broad family of intersection-closed function classes, and describe cellular free resolutions of their ideals by order complexes of the associated posets. For function classes arising from matroids, polyhedral cell complexes, and more generally interval Cohen-Macaulay posets, we show that the multigraded Betti numbers are pure, and are given combinatorially by the Möbius functions. We then apply our methods to derive bounds on the VC dimension of some important families of function classes in learning theory.

Integral closure of bipartite graph ideals

Combinatorial properties of some ideals related to bipartite graphs are examined. A description of the integral closure expressed through the log set of edge ideals of complete bipartite graphs is illustrated together with the fact that nontrivial generalized graph ideals of a strong complete quasi-bipartite graph are integrally closed.

Enumeration of Bi-Commutative–AG-groupoids

In this paper, we introduce (left, right) bi-commutative AG-groupoids and provide a simple method to test whether an arbitrary AG-groupoid is bi-commutative AG-groupoid or not. We also explore some of the general properties of these AG-groupoids. Further we introduce and study some properties of ideals in these AG-groupoids and decompose left commutative AG-groupoids by defining some congruences on these AG-groupoids

Hyper equality ideals: Basic properties

In this paper, the concept of (strong) hyper equality ideals in bounded hyper equality algebras are introduced and several properties and related results are given. Also, the properties of hyper equality ideals of the direct product of bounded hyper equality algebras are investigated; we prove that any (strong) hyper equality ideal of the direct product of hyper equality algebras is representable with respect to the product of (strong) hyper equality ideals of any direct component. In the sequel, we investigate the relationships between hyper equality ideals and hyper deductive systems in good bounded hyper equality algebras. Furthermore, we show how one can construct a hyper congruence relation via a strong hyper equality ideal so that the congruence classes form a hyper equality algebra.

Конечные циклические полукольца с полурешеточным сложением, заданным двухпорожденным идеалом натуральных чисел

В работе исследуются конечные циклические полукольца с~полурешеточным сложением, определенные как конечные циклические мультипликативные моноиды~$\langle S,\cdot \rangle$ с~введенной на них операцией сложения~$(+)$, так, что алгебраическая структура~$\langle S,+ \rangle$ является верхней полурешеткой и~выполняются законы дистрибутивности умножения относительно сложения.Описано строение конечных циклических полуколец с~полурешеточной аддитивной операцией, заданной двухпорожденным идеалом полукольца целых неотрицательных чисел.Результатом работы является теорема о строении циклических полуколец с~полурешеточной аддитивной операцией, заданной двухпорожденным идеалом полукольца целых неотрицательных чисел. Полученный результат, в частности, позволяет установить количество циклических полуколец, соответствующих каждому двухпорожденному идеалу полукольца целых неотрицательных чисел.В работе используется аппарат идеалов полукольца целых неотрицательных чисел.Получены некоторые свойства идеалов полукольца целых неотрицательных чисел, определяющих структуру конечных циклических полуколец.Работа дополняет исследования Е. М. Вечтомова и И. В. Орловой, где строение конечных циклических полуколец с~идемпотентным некоммутативным сложением описано через конечные циклические полуполя и~конечные циклические полукольца с~полурешеточным сложением.