Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

We call an ideal I of a ring R radically perfect if among all ideals whose radical is equal to the radical of I, the one with the least number of generators has this number of generators equal to the height of I. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the J _max-radical of an ideal generated by two elements. We also show that if R is a Prüfer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prüfer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prüfer domain with an extra condition imposed on it.     

Radically Perfect Prime Ideals in Polynomial Rings Over Prüfer and Pullback Rings

In this article, we study the notion of radical perfectness in Prüfer and classical pullbacks issued from valuation domains. We answer positively a question by Erdogdu of whether a domain R such that every prime ideal of the polynomial ring R[X] is radically perfect is one-dimensional. Particularly, we prove that Prüfer and pseudo-valuation domains R over which every prime ideal of the polynomial ring R[X] is radically perfect are one-dimensional domains. Moreover, the class group of such a Prüfer domain is torsion.     

Annihilator ideals,associated primes and kasch-mccoy commutative rings

If R is a ring, and A right annulet(= annihilator right ideal) then A[X] is a right annulet of the polynomial ring R[X]. (In factX can be any set of variables.). An annulet I of R[X] of this form is said to be be extended. Not all annulets of R[X] are extended, since e.g., the ascending chain on right annulets (= acc ┴) is not inherited by R[X], as, Kerr [Ke] observed. Nevertheless, maximal (minimal) annulets of a polynomial ring R[X] are extended, as a theorem of McCopy on annihilators in R[X] readily shows (see Introduction).     

Polynomials Contained in a Finite Number of Maximal Ideals

Abstract

Let R[X] be a polynomial ring in one variable over a commutative ring R. If (R,?) is a local ring then any Weierstrass polynomial in R[X] is contained only in the maximal ideal (?,X) of R[X]. We generalise this property of Weierstrass polynomials and investigate properties of polynomials contained in a finite number of maximal ideals in R[X].     

A topological characterization of the goldman prime spectrum of a commutative ring

A prime ideal p of a commutative ring R is said to be a Goldman ideal (or a G-ideal) if there exists a maximal ideal M of the polynomial ring R[X] such that p = M ∩ R. A topological space is said to be goldspectral if it is homeomorphic to the space Gold(R) of G-ideals of R (Gold(R) is considered as a subspace of the prime spectrum Spec(R) equipped with the Zariski topology). We give here a topological characterization of goldspectral spaces.     

Some Results on Finitely Laskerian Rings

In Section 1 of this note we give an example of a strongly Laskerian domain D for which the polynomial ring D[x] admits a 2-generated ideal which does not admit a primary decomposition. In Section 2 of this note we prove that if R is a quasilocal ring with M as its unique maximal ideal such that R/Ann(M) is Artinian, then for any subring T of the polynomial ring R[x], each finitely generated proper ideal of T admits a primary decomposition.     

Note on Star Operations Over Polynomial Rings

This article studies the notions of star and semistar operations over a polynomial ring. It aims at characterizing when every upper to zero in R[X] is a *-maximal ideal and when a *-maximal ideal Q of R[X] is extended from R, that is, Q = (Q ∩ R)[X] with Q ∩ R ≠0, for a given star operation of finite character * on R[X]. We also answer negatively some questions raised by Anderson–Clarke by constructing a Prüfer domain R for which the v-operation is not stable.     

Krull Dimension in Power Series Ring Over an Almost Pseudo-Valuation Domain

Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian.     

Pullbacks of Arithmetical Rings

We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.     

A remark on almost complete intersections

Let k be a perfect field and S the quotient ring of a polynomial ring k[X₁,...,X_t] with respect to a prime ideal. Let I be a prime ideal of S such that R=S/I is an almost complete intersection. Then, in his paper [2], Matsuoka proves that the homological dimension of the differential module _R/Kis infinite under the assumption that R is Cohen-Macaulay and I² is a primary idea]. In this paper we prove that the result is valid without the above assumption.     

On the Behrens radical of matrix rings and polynomial rings

It is shown that the Behrens radical of a polynomial ring, in either commuting or non-commuting indeterminates, has the form of “polynomials over an ideal”. Moreover, in the case of non-commuting indeterminates, for a given coefficient ring, the ideal does not depend on the cardinality of the set of indeterminates. However, in contrast to the Brown-McCoy radical, it can happen that the polynomial ring R[X] in an infinite set X of commuting indeterminates over a ring R is Behrens radical while the polynomial ring R〈X〉 in an infinite set Y of non-commuting indeterminates over R is not Behrens radical. This is connected with the fact that the matrix rings over Behrens radical rings need not be Behrens radical. The class of Behrens radical rings, which is closed under taking matrix rings, is described.     

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.     

The Gröbner ring conjecture in one variable

We prove that a valuation domain V has Krull dimension ≤ 1 if and only if for every finitely generated ideal I of V[X] the ideal generated by the leading terms of elements of I is also finitely generated. This proves the Gröbner ring conjecture in one variable. The proof we give is both simple and constructive. The same result is valid for semihereditary rings.     

Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

On ideals consisting entirely of zero divisors

An ideal Iin a commutative ring Ris called a z°-ideal if Iconsists of zero-divisors and for each a? Ithe intersection of all minimal prime ideals containing ais contained in I.We prove that in a large class of rings, containing Noetherian reduced rings, Zero-dimensional rings, polynomials over reduced rings and C(X), every ideal consisting of zero-divisors is contained in a prime z°-ideal. It is also shown that the classical ring of quotients of a reduced ring is regular if and only if every prime z°-ideal is a minimal prime ideal and the annihilator of a f.g. ideal consisting of zero-divisors is nonzero. We observe that z°-ideals behave nicely under contractions and extensions.     

Characterizations of schunck classes of finite groups

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a _i b _j=0 for each coefficient a _i of f and b _j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.     

ACCP Rises to the Polynomial Ring if the Ring has Only Finitely Many Associated Primes

Abstract

We show for a commutative ring R with unity: If R satisfies the ascending chain condition on principal ideals (accp) and has only finitely many associated primes, then for any set of indeterminates X the polynomial ring R[X] also satisfies accp. Further we show that accp rises to the power series ring R[[X]] if R satisfies accp and the ascending chain condition on annihilators.     

An algorithm for the divisors of monic polynomials over a commutative ring

Gilmer and Heinzer proved that given a reduced ring R, a polynomial f divides a monic polynomial in R[X] if and only if there exists a direct sum decomposition of R = R₀ ⊕ … ⊕ R_m (m ≤ deg f), associated to a fundamental system of idempotents e₀, … , e_m, such that the component of f in each R_i[X] has degree coefficient which is a unit of R_i. We propose to give an algorithm to explicitly find such a decomposition. Moreover, we extend this result to divisors of doubly monic Laurent polynomials.     

d-ideals,f d-ideals and prime ideals

Abstract

Let R be a commutative ring. An ideal I of R is called a d-ideal (f d-ideal) provided that for each a ∈ I (finite subset F of I) and b ∈ R, Ann(a) ? Ann(b) (Ann(F) ? Ann(b)) implies that b ∈ I. It is shown that, the class of z⁰-ideals (hence all sz⁰-ideals), maximal ideals in an Artinian or in a Kasch ring, annihilator ideals, and minimal prime ideals over a d-ideal are some distinguished classes of d-ideals. Furthermore, we introduce the class of f d-ideals as a subclass of d-ideals in a commutative ring R. In this regard, it is proved that the ring R is a classical ring with property (A) if and only if every maximal ideal of R is an f d-ideal. The necessary and sufficient condition for which every prime f d-ideal of a ring R being a maximal or a minimal prime ideal is given. Moreover, the rings for which their prime d-ideals are z⁰-ideals are characterized. Finally, we prove that every prime f d-ideal of a ring R is a minimal prime ideal if and only if for each a ∈ R there exists a finitely generated ideal , for some n ∈ ? such that Ann(a, I) = 0. As a consequence, every prime f d-ideal in a reduced ring R is a minimal prime ideal if and only if X= Min(R) is a compact space.