A topological prime quasi-radical

In this paper, we consider a topological prime quasi-radical μ(R), which is the intersection of closed prime ideals in a topological ring R. Examples are given that show that μ(R) is different from those topological analogs of the prime radical that have been studied earlier. The topological prime quasi-radicals of matrix rings and rings of polynomials are investigated. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 3, pp. 11–22, 2004.     

Block separation in solvable groups

If π is a set of primes, a finite group G is block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there exists a prime p ∈ π such that α and β lie in different Brauer p-blocks. A group G is block separated if it is separated by the set of prime divisors of |G|. Given a set π with n different primes, we construct an example of a solvable π-group G which is block separated but it is not separated by every proper subset of π. Received: 22 December 2004     

Remarks on ��-strongly irreducible ideals

In this article, we study α-irreducible and α-strongly irreducible ideals of a commutative ring. The relations between α-strongly irreducible ideals of a ring and α-strongly irreducible ideals of localization of the ring are also studied.     

The Jacobson radical of the Laurent series ring

For a large class of rings A, including all rings with right Krull dimension, it is proved that for every automorphism ϕ of the ring A, the Jacobson radical of the skew Laurent series ring A((x, ϕ)) is nilpotent and coincides with N((x, ϕ)), where N is the prime radical of the ring A. If A/N is a ring of bounded index, then the Jacobson radical of the Laurent series ring A((x)) coincides with N((x)). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 209–215, 2006.     

On Andrunakievich’s chain and Koethe’s problem

In 1969 Andrunakievich asked whether one gets a ring without nonzero nil left ideals from an arbitrary ring R by factoring out the ideal A(R) which is the sum of all nil left ideals of R. Recently, it was shown that this problem is equivalent to Koethe’s problem. In this context one may consider the chain of ideals, which starts with A ₁(R) = A(R) ⊆ A ₂(R), where A ₂(R)/A ₁(R) = A(R/A ₁(R)), and extends by repeating this process. We study the properties of this chain and show that, assuming a negative solution of Koethe’s problem, this chain can terminate at any given ordinal number.     

Differential rings and ore extensions: Brown-McCoy rings

We consider here a ringK, a derivationD ofK and the differential polynomial ringR=K[X;D]. The ringK is said to be a Brown-McCoy ring if the prime radical coincides with the Brown-McCoy radical in every homomorphic image ofK. AD-Brown-McCoy ring is defined in a similar way. We prove the following conditions are equivalent: (i)K is aD-Brown-McCoy ring; (ii)R is a Brown-McCoy ring and for every maximal idealM ofR,K/(MνK) is aD-simple ring with 1. In addition, we give some applications and examples on the study of the transfer of the property of being a Brown-McCoy ring betweenK andR. Further, we study the relation between the prime and theD-prime ideals of a differential intermediate extension of a liberal extension. This paper was supported by a fellowship awarded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.     

ASSOCIATIONS AND VALUATIONS

Abstract

Right cones are semigroups for which the lattice of right ideals is a chain and a left cancellation law holds; valuation rings, the cones of ordered groups, and initial segments of ordinal numbers are examples. Two such cones are associated if they have isoniorphic lattices of right ideals so that ideals, prime ideals, and completely prime ideals correspond to each other. A list of problems is discussed. In Proposition 3.11 it is proved that the canonical mapping from a right invariant right chain domain R onto the associated right holoid can be extended to a valuation from the skew field Q(R) of quotients of R onto an ordered group if and only if Ja ? aJ for all a ∈ R and J = J(R), the Jacobson radical of R.     

Finitely Presented Modules Over Semihereditary Rings

We prove that each almost local-global semihereditary ring R has the stacked bases property and is almost Bézout. More precisely, if M is a finitely presented module, its torsion part tM is a direct sum of cyclic modules where the family of annihilators is an ascending chain of invertible ideals. These ideals are invariants of M. Moreover, M/tM is a projective module which is isomorphic to a direct sum of finitely generated ideals. These ideals allow us to define a finitely generated ideal whose isomorphism class is an invariant of M. The idempotents and the positive integers defined by the rank of M/tM are invariants of M too. It follows that each semihereditary ring of Krull-dimension one or of finite character, in particular each hereditary ring, has the stacked base property. These results were already proved for Prüfer domains by Brewer, Katz, Klinger, Levy, and Ullery. It is also shown that every semihereditary Bézout ring of countable character is an elementary divisor ring.     

Prime ideals in restricted differential operator rings

In this paper restricted differential operator rings are studied. A restricted differential operator ring is an extension of ak-algebraR by the restricted enveloping algebra of a restricted Lie algebra g which acts onR. This is an example of a smash productR #H whereH=u (g). We actually deal with a more general twisted construction denoted byR * g where the restricted Lie algebra g is not necessarily embedded isomorphically inR * g. Assume that g is finite dimensional abelian. The principal result obtained is Incomparability, which states that prime idealsP ₁ ⊆P ₂ ⊂R * g have different intersections withR. We also study minimal prime ideals ofR * g whenR is g-prime, showing that the minimal primes are precisely those having trivial intersection withR, that these primes are finite in number, and their intersection is a nilpotent ideal. Prime and primitive ranks are considered as an application of the foregoing results.     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

非结合非分配的环(Ⅱ)

本文继上文(Ⅰ)的理论,共分二节,第一节继续上文的理想理论,引进两非环的半素理想及半准素理想概念,并对它们作基本性质的研究.第二节引进根及半单纯概念建立分解成单纯子环直和的有关诸定理.     

The abc-conjecture for Algebraic Numbers

The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.     

Orders of accumulation of entropy on manifolds

For a continuous self-map T of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold M and any countable ordinal α, we construct a continuous, surjective self-map ofM having order of accumulation of entropy α. If the dimension of M is at least 2, then the map can be chosen to be a homeomorphism.     

GENERALISED IDEALS OF RINGS

Abstract

Using a general definition of a regularity for rings, F- and F- qausi-ideals of a ring are defined. These concepts are shown to be generalizations of ideals or one-sided ideals of a ring. An F-semi prime F—(F-quasi-) ideal of a ring R is also defined. F-regular rings are characterized in terms of F-semi prime F- (F-quasi-) ideals for a large class of polynomial regularities including some well known regularities. A more general characterization of the prime radical β(R) of a ring are given in terms of F—(F-quasi-) ideals.     

Spectral partially ordered sets

We examine some topics related to (gold)spectral partially ordered sets, i.e., those that are order isomorphic to (Goldman) prime spectra of commutative rings. Among other results, we characterize the partially ordered sets that are isomorphic to prime spectra of rings satisfying the descending chain condition on radical ideals, and we show that a dual of a tree is isomorphic to the Goldman prime spectrum of a ring if and only if every chain has an upper bound. We also give some new methods for constructing (gold)spectral partially ordered sets.     

The intersection of the powers of the radical in Noetherian P.I. rings

LetR be a ring and J its radical. DefineJ ₁=∩Jⁿ, J₂=∩J ₁ ⁿ ,…,… J_k=∩J _k−1 ⁿ .... It is shown that in a ringR satisfying a polynomial identity and the ascending chain condition on ideals,J _k =0 for some appropriatek. The work of the first author was supported by an NSF grant at the University of Chicago. The work of the second author was supported by an NSF grant at the University of California, San Diego.     

Rings of quotients of f-rings by gabriel filters of ideals

The article examines the role of Gabriel filters of ideals in the ontext of semiprime f-rings. It is shown that for every 2-convex semiprime f-ring Aand every multiplicative filter B of dense ideals the ring of quotients of A by B, namely the direct limit of the Hom _A (I, A) over all I∈ B, is an l-subring of QA, the maximum ring of quotients. Relative to the category of all commutative rings with identity, it is shown that for every 2-convex semiprime f-ring A qA, the classical ring of quotients, is the largest flat epimorphic extension of A. If Ais also a Prüfer ring then it follows that every extension of Ain qA is of the form S ^-1A for a suitable multiplicative subset S. The paper also examines when a Utumi ring of quotients of a semiprime f-ring is obtained from a Gabriel filter. For a ring of continuous functions C(X), with Xcompact, this is so for each C(U) and C ^*(U), when Uis dense open, but not for an arbitrary direct limit of C(U),taken over a filter base of dense open sets. In conclusion, it is shown that, for a complemented semiprime f-ring A, the ideals of Awhich are torsion radicals with respect to some hereditary torsion theory are precisely the intersections of minimal prime ideals of A.     

Prime-representing functions

We construct prime-representing functions. In particular we show that there exist real numbers α > 1 such that ⌈α ^{2 n} ⌉ is prime for all n ∈ ℕ. Indeed the set consisting of such numbers α has the cardinality of the continuum.     

Rings with divisibility on chains of ideals

We present a generalization of the ascending and descending chain condition on one-sided ideals by means of divisibility on chains. We say that a ring R satisfies ACC_d on right ideals if in every ascending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is a left multiple of the following one; that is, each right ideal in the chain, except for a finite number, is divisible by the following one. We study these rings and prove some results about them. Dually, we say that a ring R satisfies DCC_d on right ideals if in every descending chain of right ideals of R, each right ideal in the chain, except for a finite number of right ideals, is divisible by the previous one. We study these conditions on rings, in general and in special cases.