Power Series Rings Satisfying a Zero Divisor Property

In this note we continue to study zero divisors in power series rings and polynomial rings over general noncommutative rings. We first construct Armendariz rings which are not power-serieswise Armendariz, and find various properties of (power-serieswise) Armendariz rings. We show that for a semiprime power-serieswise Armendariz (so reduced) ring R with a.c.c. on annihilator ideals, R[[x]] (the power series ring with an indeterminate x over R) has finitely many minimal prime ideals, say B ₁,…,B _m , such that B ₁… B _m  = 0 and B _i  = A _i [[x]] for some minimal prime ideal A _i of R for all i, where A ₁,…,A _m are all minimal prime ideals of R. We also prove that the power-serieswise Armendarizness is preserved by the polynomial ring extension as the Armendarizness, and construct various types of (power-serieswise) Armendariz rings.     

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.     

Irreducibility in the Total Ring of Quotients

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.     

A basic functional identity with applications to Jordan σ-biderivations

Let R be a noncommutative prime ring with extended centroid C and maximal left ring of quotients Q_ml(R). The aim of the paper is to study a basic functional identity concerning bi-additive maps on R. Precisely, it is proved that a bi-additive map B:R×R→Q_ml(R) satisfying [B(x,y),[x,y]] = 0 for all x,y∈R must be of the form (x,y)?λ[x,y]+μ(x,y) for x,y∈R, where λ∈C and μ:R×R→C is a bi-additive map. As applications to the theorem, Jordan σ-biderivations with σ an epimorphism and additive commuting maps on noncommutative Lie ideals of R are characterized.     

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.     

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.     

Valuation ideals of order one in 2-dimensional regular local rings

Let υ be a prime divisor of a 2-dimensional regular local ring (R m) with algebraically closed residue field k. Zariski showed that a prime divisor υ of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple υ-ideals including I, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements.Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.     

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec _r (R) (resp. Spec _l (R), Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec _r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec _r (R) is a normal space if and only if Spec _l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.     

Partial cancellation of modules

We investigate partial cancellation of modules and show that if an ideal I of an exchange ring R has stable range one, then A⊕B≌A⊕C implies B≌C for all A∈FP (I). The converse is true when R is a regular ring. For an ideal I of a regular ring, we also show that I has stable range one if and only if perspectivity is transitive in L(A) for all A∈ FP (I). These give nontrivial generalizations for unit-regularity. This revised version was published online in June 2006 with corrections to the Cover Date.     

Universal Lying-Over Rings

A (commutative unital) ring R is said to satisfy universal lying-over (ULO) if each injective ring homomorphism R → T satisfies the lying-over property. If R satisfies ULO, then R = tq(R), the total quotient ring of R. If a reduced ring satisfies ULO, it also satisfies Property A. If a ring R = tq(R) satisfies Property A and each nonminimal prime ideal of R is an intersection of maximal ideals, R satisfies ULO. If 0 ≤ n ≤ ∞, there exists a reduced (resp., nonreduced) n-dimensional ring satisfying ULO. The A + B construction is used to show that if 2 ≤ n < ∞, there exists an n-dimensional reduced ring R such that R = tq(R), R satisfies Property A, but R does not satisfy ULO.     

PRIME IDEALS IN RINGS WITH INVOLUTION

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.     

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.     

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.     

Presimplifiable and Cyclic Group Rings

Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(C _n ) are presimplifiable if and only if n = p ^m , p ∈ J(R), p is prime, and R is presimplifiable. We show the necessity of n = p ^m , and we prove the sufficiency when n = 2, 3, 4. These results were made possible by a new formula derived herein for the circulant determinantal coefficients.     

A result concerning nilpotent values with generalized skew derivations on Lie ideals

ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)ⁿ = 0 for all x,y∈L, then char(R) = 2 and R?M₂(C), the ring of 2×2 matrices over C.     

Constants of Derivation and Differential Ideals

ABSTRACT

Let R be a differential domain finitely generated over a differential field F of characteristic 0. Let C be the subfield of differential constants of F. This paper investigates conditions on differential ideals of R that are necessary or sufficient to guarantee that C is also the set of constants of differentiation of the quotient field, E, of R. In particular, when C is algebraically closed and R has a finite number of height one differential prime ideals, there are no new constants in E. An example where F is infinitely generated over C shows the converse is false. If F is finitely generated over C and R is a polynomial ring over F, sufficient conditions on F are given so that no new constants in E does imply only finitely many height one prime differential ideals in R. In particular, F can be Q¯(T) where T is a finite transcendence set.

     

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).      

Space of minimal prime ideals of a ring without nilpotent elements

Hull-kernel topology on the set ∑(R) of prime ideals of a ring R with unity and without nilpotent elements is discussed. The restriction of this topology to the set π(R) of minimal prime ideals of R has been investigated in detail. The compactness of π(R) has been characterized in several ways. An interesting characterization of Baer rings is given.A functorial correspondence between the category of rings having the property that every prime ideal contains a unique minimal prime ideal and their minimal spectra is established.