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.     

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.     

Local bijective gabriel correspondence and relative fully boundedness in τ-noetherian rings

Let Rbe a right τ-noetherian ring, where τ denotes a hereditary torsion theory on the category of right R-modules. It is shown that every essential τ-closed right ideal of every prime homomorphic image of Rcontains a nonzero two-sided ideal if and only if any two τ-torsionfree injective indecomposable right R-modules with identical associated prime ideals are isomorphic, and for any τ-closed prime ideal Pthe annhilator of a finitely generated P-tame right R-module cannot be a prime ideal properly contained in P. Furthermore, if in the last condition finitely generated is replaced by r-noetherian, then all τ-noetherian τ-torsionfree modules turn out to be finitely annihilated.     

On finiteness of multiplication modules

Our main aim in this note, is a further generalization of a result due to D. D. Anderson, i.e., it is shown that if R is a commutative ring, and M a multiplication R-module, such that every prime ideal minimal over Ann (M) is finitely generated, then M contains only a finite number of minimal prime submodules. This immediately yields that if P is a projective ideal of R, such that every prime ideal minimal over Ann (P) is finitely generated, then P is finitely generated. Furthermore, it is established that if M is a multiplication R-module in which every minimal prime submodule is finitely generated, then R contains only a finite number of prime ideals minimal over Ann (M).      

Differential Calculi on Quantum Groupoids

Let R be a commutative ring with Noetherian spectrum in which zero is a primary ideal. We determine the minimal zero-dimensional extensions of R when every regular prime ideal of R is contained in only finitely many prime ideals. This extends previous results of the first author for dim (R) ≤1. We also present a characterization of the partially ordered set of prime ideals in a ring with Noetherian spectrum.     

M-Canonical ideals in integral domains

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R     

On Local ☆-Completely Integrally Closed Domains

Let ☆ be a star operation on an integral domain R. The domain R is a ☆-CICD if (AA ^?1)^☆ = R for all nonzero (fractional) ideals A of R. In this article, we prove that, if the maximal ideal of a local ☆-CICD is a ☆-ideal, then R is ☆-principal ideal domain. We also establish that any ☆-CICD R is locally a PID when ☆ is induced by the localizations at prime ideals of R.     

Valuation Domains Whose Products of Free Modules Are Separable

It is proved that if R is a valuation domain with maximal ideal P and if R _L is countably generated for each prime ideal L, then R ^R is separable if and only R _J is maximal, where J = ∩ _n∈? P ⁿ .     

Finite character of finitely stable domains

A commutative domain is finitely stable if every nonzero finitely generated ideal is stable, i.e. invertible over its endomorphism ring. A domain satisfies the local stability property provided that every locally stable ideal is stable.We prove that a finitely stable domain satisfies the local stability property if and only if it has finite character, that is every nonzero ideal is contained in at most finitely many maximal ideals. This result allows us to answer the open problem of whether every Clifford regular domain is of finite character.     

The Rees Valuations of Complete Ideals in a Regular Local Ring

Let I be a complete m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. In the case of dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points Lipman proves a unique factorization involving special *-simple complete ideals and possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and R/m = T/m _T . We prove that the special *-simple complete ideal P _RT has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transformations from R to T. We also examine conditions for a complete ideal to be projectively full.     

Localization of Injective Modules Over Arithmetical Rings

It is proved that localizations of injective R-modules of finite Goldie dimension are injective if R is an arithmetical ring satisfying the following condition: for every maximal ideal P, R _P is either coherent or not semicoherent. If, in addition, each finitely generated R-module has finite Goldie dimension, then localizations of finitely injective R-modules are finitely injective too. Moreover, if R is a Prüfer domain of finite character, localizations of injective R-modules are injective.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Finiteness and Skolem Closure of Ideals for Non-Unibranched Domains

A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 _{k, a}  = {f ∈ Int(D) | f(a) ∈ 𝔪 ^k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 _a  = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated.     

On Finite Saturated Chains of Overrings

If a domain R, with quotient field K, has a finite saturated chain of overrings from R to K, then the integral closure of R is a Prüfer domain. An integrally closed domain R with quotient field K has a finite saturated chain of overrings from R to K with length m ≥ 1 iff R is a Prüfer domain and |Spec(R)| =m + 1. In particular, we prove that a domain R has a finite saturated chain of overrings from R to K with length dim(R) iff R is a valuation domain and that an integrally closed domain R has a finite saturated chain of overrings from R to K with length dim (R) +1 iff R is a Prüfer domain with exactly two maximal ideals such that at most one of them fails to contain every non-maximal prime. The relationship with maximal non-valuation subrings is also established.     

Minimal prime ideals of reduced rings

Let R be a reduced ring with Q its Martindale symmetric ring of quotients, and let B be the complete Boolean algebra of all idempotents in C, where C is the extended centroid of R. It is proved that every minimal prime ideal of R must be of the form mQ∩R for some maximal ideal m of B but the converse is in general not true. In addition, if R is centrally closed or has only finitely many minimal prime ideals, then the converse also holds. By applying the explicit expression, many properties of minimal prime ideals of reduced rings are realized more easily.     

Quotient rings of finitely generated P.I. Algebras

This paper generalizes properties which hold for localization of Azumaya algebras, in two directions. Firstly, fully left bounded left Noetherian rings, especially finitely generated Noetherian algebras, are considered. It is noted that for such rings every idempotent kernel functor a is symmetric, i.e. the filter T(σ) of a-dense left ideals has a basis of a-dense ideals. A prime ideal P of a f.l.b.l.N. ring R is localizable if and only if it is the intersection of the P-critical left ideals. In case R is a finitely generated algebra over its (Noetherian) center C, we apply the technique of “descent” of kernel functors. If a is a symmetric kernel functor such that R(A n c) S T(σ) for every A G T(σ) and such that a has property (T) then there is a kernel functor a’ on C-modules such that Qσ (R) ?Q? ,(R). If P is a prime ideal of R then σ_- descends to C if and only if P is localizable. Secondly, a class of rings is described in terms of the Zariski topology on Spec. The imposed condition is weaker than maximal centrallity and does not imply fully left boundedness either, but the good properties of Spec R in case R is an Azumaya algebra are preserved.     

On Lie Nilpotent Rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z _n (R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring ?Z _n (R) ∪ C? of R generated by the subset Z _n (R) ∪ C of R is also Lie nilpotent of index n.     

Forcing Linearity Numbers for Multiplication Modules

In this article, we prove that for any multiplication module M, the forcing linearity number of M, fln(M), belongs to {0,1,2}, and if M is finitely generated whose annihilator is contained in only finitely many maximal ideals, then fln(M) = 0. Also, the forcing linearity numbers of multiplication modules over some special rings are given. We also show that every multiplication module is semi-endomorphal.