On Standard Ideals

In the paper,we study a class of standard ideals which are more general than the m-primary standared ideals discussed in[2].We will prove an important equality concerning I-weak sequences;thus a generalization of the equality of [2] is established.     

Commutative ideal theory without finiteness conditions: Completely irreducible ideals

An ideal of a ring is completely irreducible if it is not the intersection of any set of proper overideals. We investigate the structure of completely irrreducible ideals in a commutative ring without finiteness conditions. It is known that every ideal of a ring is an intersection of completely irreducible ideals. We characterize in several ways those ideals that admit a representation as an irredundant intersection of completely irreducible ideals, and we study the question of uniqueness of such representations. We characterize those commutative rings in which every ideal is an irredundant intersection of completely irreducible ideals.

     

UNIQUE FACTORIZATION AND BIRTH OF ALMOST PRIMES

ABSTRACT

In studying unique factorization of domains we encountered a property of ideals. Using that we define the notion of almost prime ideals and prove that in Noetherian domains almost prime ideals are primary. We also prove that in a regular domain almost primes are precisely primes. Further, we define strictly nonprime ideals and study some inter relations between almost prime ideals, strictly nonprime ideals and factorization of ideals.     

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.     

On the S 2-Fication of Some Toric Varieties

In this article we prove the following:

1. Some results on the Cohen–Macaulayness of the canonical module;

2. We study the S ₂-fication of rings which are quotients by lattices ideals;

3. Given a simplicial lattice ideal of codimension two I, its Macaulayfication is given explicitly from a system of generators of I.

     

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.     

Licci ideals and the non-almost complete intersection locus

We extend results of Huneke and Ulrich on the structure of ideals in the linkage class of a complete intersection, for the class of ideals which are linked to a complete intersection in an even number of steps. In particular for such ideals the non-almost complete intersection locus has codimension at most ten.

     

Ideals of Finite-Dimensional Pointed Hopf Algebras of Rank One

Let H be a finite-dimensional pointed Hopf algebra of rank one over an algebraically closed field of characteristic zero.In this paper we show that any finite-dimensional indecomposable H-module is generated by one element.In particular,any indecomposable submodule of H under the adjoint action is generated by a special element of H.Using this result,we show that the Hopf algebra H is a principal ideal ring,i.e.,any two-sided ideal of H is generated by one element.As an application,we give explicitly the generators of ideals,primitive ideals,maximal ideals and completely prime ideals of the Taft algebras.     

Quasi-Schreier Domains II

We study a class of integral domains characterized by the property that every nonzero finite intersection of principal ideals is a directed union of invertible ideals.     

关于循环环的理想、素理想和极大理想

研究了循环环R=的理想、素理想和极大理想的个数和结构,得到了如下结论:1)理想:(1)若|R|=∞,则R共有无穷多个理想:;(2)若|R|=n,设n的正因数个数为T(n),则R共有T(n)个理想:.2)素理想:(1)若|R|=∞,设a^2=ka(k≥0),①当k=0时,R的素理想只有R;②当k>0时,R的素理想共有无穷多个,它们是:{0}、R及;(2)若|R|=n>1,设a^2=ka,0≤k.3)极大理想:(1)若|R|=∞,则R有无限多个极大理想,它们是;(2)若|R|=n>1,设n的互不相同的素因数个数为ψ(n),则R共有ψ(n)个极大理想:(pa|p是n的素因数).     

Ideals of Some Matrix Rings

For a commutative ring K the conception of a strongly maximal ideal J was introduced by Kuzucuoglu and Levchuk in 2000. Denote by R_n(K,J) the ring of all n×n-matrices over K with elements from J on and above the main diagonal. Recent results on ideals of the ring R_n(K,J) for this case, ideals of the associated Lie ring and normal subgroups of the adjoint group are considered in this paper. Also ideals of R_n(K,J) for the case of an arbitrary associative ring K with the identity are investigated.     

PP-理想

引入了环的PP-理想的概念,证明了PP-理想的一些性质及它与一些环、模、理想的联系,得到了一些有用的结论.     

Reflexivity with maximal ideal axes

The reflexive property for ideals was introduced by Mason and has important roles in noncommutative ring theory. We in this note study rings with the reflexivity whose axis is given by maximal ideals (simply, an RM ring) which are a generalization of symmetric rings. It is first shown that the reflexivity of a ring and the RM ring property are independent of each other, noting that both of them are generalizations of ideal-symmetric rings. We connect RM rings with reflexive rings in various situations raised naturally in the procedure. As a generalization of RM rings, we also study the structure of the reflexivity with the maximal ideal axis on idempotents (simply, an RMI ring) and then investigate the structure of minimal non-Abelian RMI rings (with or without identity) up to isomorphism.     

Divisorial Linear Algebra of Normal Semigroup Rings

We investigate the minimal number of generators and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that (I)C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of and depth in arithmetic progressions in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.     

Similarity invariants for matrices over a commutative Artinian chain ring

Suppose that R is a commutative Artinian chain ring, A is an m × m matrix over R, and S is a discrete valuation ring such that R is a homomorphic image of S. We consider m ideals in the polynomial ring over S that are similarity invariants for matrices over R, i.e., these ideals coincide for similar matrices. It is shown that the new invariants are stronger than the Fitting invariants, and that new invariants solve the similarity problem for 2 × 2 matrices over R.     

COMMUTATIVE RINGS WITH FINITE QUOTIENT FIELDS

Abstract

We consider the class of all commutative reduced rings for which there exists a finite subset T ? A such that all projections on quotients by prime ideals of A are surjective when restricted to T. A complete structure theorem is given for this class of rings,and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals.     

Rings whose singular ideals are nil

Abstract

It is well known that when a ring R satisfies ACC on right annihilators of elements, then the right singular ideal of R is nil, in this case, we say R is right nil-singular. Many classes of rings whose singular ideals are nil, but do not satisfy the ACC on right annihilators, are presented and the behavior of them is investigated with respect to various constructions, in particular skew polynomial rings and triangular matrix rings. The class of right nil-singular rings contains π-regular rings and is closed under direct sums. Examples are provided to explain and delimit our results.