On the Factorial Closure of Certain Monomial Modules

Abstract

Let R = K[y ₁,…,y _t ] be an affine domain over a field K and I be a nonzero proper ideal of R. In Sec. 1 of this note, we characterize when (K + I, R) is a Mori pair. In Sec. 2 of this note, we prove the following theorem: Let A ? B be domains such that C/Q is Mori for each subring C of B containing A and for any prime ideal Q of C. Then dim A ? 1 ≤ dim B ≤ dim A + 1 and if dim A > 1 or dim B > 1 then dim A = dim B.     

On the action of derivations on nilpotent ideals of associative algebras

Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F.     

The set of indeterminate rings of a normal pair as a partially ordered set

Let R ì S{R\subset S} be an extension of integral domains and let [R, S] be the set of intermediate rings between R and S ordered by inclusion. If (R, S) is normal pair and [R, S] is finite, we do prove that there exists a semi-local Prüfer ring T with quotient field K such that [R,S] @ [T,K]{[R,S]\cong \lbrack T,K]} (as a partially ordered set). Consequently, any problem relative to the finiteness conditions in [R, S] can be investigated in the particular case where R is a semi-local Prüfer ring with quotient field S.     

Zero-divisor graphs of amalgamated duplication of a ring along an ideal

Let R be a commutative ring with identity and let I be an ideal of R. Let R?I be the subring of R×R consisting of the elements (r,r+i) for r∈R and i∈I. We study the diameter and girth of the zero-divisor graph of the ring R?I.     

MULTIPLICATIVELY CLOSED SETS OF IDEALS AND RESIDUAL DIVISION

Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I _Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A ^Γ = ∪ {A : _T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I _Γ is an ideal in R, I → I _Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R ^Γ. Also, A ^Γ is an R-algebra and A → A ^Γ is a closure operation on any set A = {A; A is an R-algebra, R ? A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I _Γ and (IA)_ΓA and between the R-algebras R ^Γ and A ^Γ.

     

Irreducibility and Reduction Number of Ideals in a One Dimensional Analytically Irreducible Ring

Let R be a one dimensional analytically irreducible ring, and let I be an integral ideal of R. We study the irreducibility and the reduction number of I in relation with the corresponding semigroup ideal v(I) in v(R), where v(R) is the semigroup of values of R. It turns out that, if v(I) is irreducible, then I is irreducible, but the converse does not hold in general. We show also that the reduction number of I in R can assume any positive value less than the multiplicity of R and can be different from the reduction number of v(I).     

A Note on Conductor Ideals

Let S be a commutative ring with identity and R a unitary subring of S. An ideal I of S is called an R-conductor ideal of S if I = {x ∈ S | xS ? V} for some intermediate ring V of R and S. In this note, we present necessary and sufficient criteria for being an R-conductor ideal of S. We generalize several well known facts about them and present a simple approach to rediscover the results of both old and recent articles. We sketch the boundaries of our criteria by providing a few counterexamples.     

On the Prime Ideal Structure of Bhargava Rings

Let D be an integral domain with quotient field K. A Bhargava ring over D is defined to be 𝔹 _x (D): = {f ∈ K[X] | ? a ∈ D, f(xX + a) ∈ D[X]}, where x ∈ D. A Bhargava ring over D is a subring of the ring of integer-valued polynomials over D. In this article, we study the prime ideal structure and calculate the Krull and valuative dimension of Bhargava rings over a general domain D.     

Derivations on ideals in commutative AW*-algebras

LetA be a commutativeAW*-algebra.We denote by S(A) the *-algebra of measurable operators that are affiliated with A. For an ideal I in A, let s(I) denote the support of I. Let Y be a solid linear subspace in S(A). We find necessary and sufficient conditions for existence of nonzero band preserving derivations from I to Y. We prove that no nonzero band preserving derivation from I to Y exists if either Y ? Aor Y is a quasi-normed solid space. We also show that a nonzero band preserving derivation from I to S(A) exists if and only if the boolean algebra of projections in the AW*-algebra s(I)A is not σ-distributive.     

PrÜfer-Like Domains and the Nagata Ring of Integral Domains

A subring A of a Prüfer domain B is a globalized pseudo-valuation domain (GPVD) if (i) A?B is a unibranched extension and (ii) there exists a nonzero radical ideal I, common to A and B such that each prime ideal of A (resp., B) containing I is maximal in A (resp., B). Let D be an integral domain, X be an indeterminate over D, c(f) be the ideal of D generated by the coefficients of a polynomial f ∈ D[X], N = {f ∈ D[X] | c(f) = D}, and N _v  = {f ∈ D[X] | c(f)^?1 = D}. In this article, we study when the Nagata ring D[X] _N (more generally, D[X] _{N v} ) is a GPVD. To do this, we first use the so-called t-operation to introduce the notion of t-globalized pseudo-valuation domains (t-GPVDs). We then prove that D[X] _{N v} is a GPVD if and only if D is a t-GPVD and D[X] _{N v} has Prüfer integral closure, if and only if D[X] is a t-GPVD, if and only if each overring of D[X] _{N v} is a GPVD. As a corollary, we have that D[X] _N is a GPVD if and only if D is a GPVD and D has Prüfer integral closure. We also give several examples of integral domains D such that D[X] _{N v} is a GPVD.     

Valuation and Pseudovaluation Subrings of an Integral Domain

Let R, S be two rings. We say that R is a valuation subring of S (R is a VD in S, for short) if R is a proper subring of S and whenever x ∈ S, we have x ∈ R or x ^?1 ∈ R. We denote by Nu(R) the set of all nonunit elements of a ring R. We say that R is a pseudovaluation subring of S (R is a PV in S, for short) if R is a proper subring of S and x ^?1 a ∈ R, for each x ∈ S?R, a ∈ Nu(R). This article deals with the study of valuation subrings and pseudovaluation subrings of a ring; interactions between the two notions are also given. Let R be a PV in S; the Krull dimension of the polynomial ring on n indetrminates over R is also computed.     

Symplectic modules over overrings of polynomial rings

Let B be a commutative Noetherian ring of dimension d and let S be a set of all monic polynomials in B[X]. Let A be a subring of S ⁻¹ B[X] which contains B[X]. Let P be a symplectic A-module of rank 2n ≥ d, n > 0. Then we prove that ESp (A ² ⊥ P, 〈,〉) acts transitively on Um (A ² ⊕ P).     

Cohen-Macaulayness of Special Fiber Rings

Abstract

Let (R, 𝔪) be a Noetherian local ring and let Ibe an R-ideal. Inspired by the work of Hübl and Huneke, we look for conditions that guarantee the Cohen-Macaulayness of the special fiber ring ? = ?/𝔪? of I, where ? denotes the Rees algebra of I. Our key idea is to require ‘good’ intersection properties as well as ‘few’ homogeneous generating relations in low degrees. In particular, if Iis a strongly Cohen-Macaulay R-ideal with G _?and the expected reduction number, we conclude that ? is always Cohen-Macaulay. We also obtain a characterization of the Cohen-Macaulayness of ?/K? for any 𝔪-primary ideal K. This result recovers a well-known criterion of Valabrega and Valla whenever K = I. Furthermore, we study the relationship between the Cohen-Macaulay property of the special fiber ring ? and the Cohen-Macaulay property of the Rees algebra ? and the associated graded ring 𝒢 of I. Finally, we focus on the integral closedness of 𝔪I. The latter question is motivated by the theory of evolutions.     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

The Inclusion Ideal Graph of Rings

Let R be a ring with unity. The inclusion ideal graph of a ring R, denoted by In(R), is a graph whose vertices are all nontrivial left ideals of R and two distinct left ideals I and J are adjacent if and only if I ? J or J ? I. In this paper, we show that In(R) is not connected if and only if R ? M ₂(D) or D ₁ × D ₂, for some division rings, D, D ₁ and D ₂. Moreover, we prove that if In(R) is connected, then diam(In(R)) ≤3. It is shown that if In(R) is a tree, then In(R) is a caterpillar with diam(In(R)) ≤3. Also, we prove that the girth of In(R) belongs to the set {3, 6, ∞}. Finally, we determine the clique number and the chromatic number of the inclusion ideal graph for some classes of rings.     

Note on ideal-transforms, Rees rings, and Krull rings

Let I be an ideal in a Noetherian ring R and let T(I) be the ideal-transform of R with respect to I. Several necessary and sufficient conditions are given for T(I) to be Noetherian for a height one ideal I in an important class of altitude two local domains, and some specific examples are given to show that the integral closure T(I)′ and the complete integral closure T(I)″ of T(I) may differ, even when R is an altitude two Cohen-Macaulay local domain whose integral closure is a regular domain and a finite R-module. It is then shown that T(I)″ is always a Krull ring, and if the integral closure of R is a finite R-module, then T(I)′ is contained in a finite T(I)-module. Finally, these last two results are applied to certain symbolic Rees rings.     

Realizing Sets of Associated Prime Ideals

Let R be a Noetherian ring, and let S be a finite subset of Spec R. We characterize when there exists an ideal I such that S equals the set of associated prime divisors of I.     

Resolutions of monomial almost complete intersections and generalizations

Let S=K[x₁,…,x_n] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.