CHARACTERIZING THE NIL RADICAL OF A GAMMA RING

ABSTRACT

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.     

A NOTE ON THE g-PRIME RADICAL IN GAMMA RINGS

ABSTRACT

The g-prime radical of a Γ-ring M is equal to either the zero ideal or the prime radical of M. If the prime radical of M is a non-zero ideal, then the following three conditions are equivalent; (i) g-prime radical of M is equal to the prime radical of M; (ii) every g-prime ideal is a prime ideal; and (iii) every g-semiprime ideal is a semiprime ideal.     

EQUIPRIME Γ-NEAR-RINGS

Abstract

Equiprime and strongly equiprime near-rings were recently defined by the present authors, together with S. Veldsman. In the present paper, the concepts are introduced for Γ-near-rings, and give rise to Kurosh- Amitsur radicals. If M is a Γ-near-ring and L is its left operator near-ring, then R(L)⁺ = R(M), where R(—) in both cases denotes either the equiprime or the strongly equiprime radical.     

RADICALS IN GENERAL Γ-NEAR-RINGS

Abstract

The J ₂ and J ₃ radicals for zerosymmetric Γ-near-rings were recently defined by the author. In the present paper we define the J ₂₍₀₎ and J ₃₍₀₎ radicals for arbitrary Γ-near-rings. These radicals are sirmlar to corresponding ones which were recently defined by Veldsman for near-rings. Let M be a r-near-ring with left operator near-ring L. Then J _κ(0)(L)⁺ = J _{κ (0)} (M), k. = 2,3. If A is an ideal of M, then J _{κ (0)} (A) ? J _{κ (o)}(M) ∩ A, with equality when k = 3 and A is left invariant. J ₃₍₀₎ is a Kurosh-Amitsur radical in the variety of Γ-near-rings.     

A BROWN-McCOY RADICAL FOR T-RINGS

Abstract

A notion of G-regularity is introduced for a Γ-ring M, and from this notion a Brown-McCoy radical, B(M) is defined. B(M) is shown to be a radical in the sense of Kurosh and Amitsur, and analogies of various well-known results on the Brown-McCoy radical of a ring are proved. A “right” Brown McCoy radical, B'(M) can also be defined. In general, B(M) ? B'(M).     

所有同态象其Jacobson根与单根相同的Γ-环

设M是Nobusawa意义下的Г-环,S.Kyuno定义了环M_2=其中R,L分别是M的右、左算子环.本文首先刻画了环M_2的本原理想与Ja-cobson根.其次引进了一类新的Г-环称为PM Г-环,建立了Г-环M、矩阵Г_(n,m)-环M_(m,n)、Г-环M的右(左)算子环R(L)、M-环Г及M_2的PM性质之间的关系.最后,给出了Г-环一般形式的Jacobson性质,Jacobson性质、Brown-McCoy性质以及PM性质为其特殊情况.     

CLASSICAL QUOTIENT RINGS OF GROUP RINGS

Abstract

Throughout G will denote a free Abelian group and Z(R) the right singular ideal of a ring R. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R/Z(R) is semiprime, Z(R) is rationally closed, and Z(R) contains no closed uniform right ideals. We prove that R is a Cl-ring if and only if the group ring RG is a C1-ring. If RG has the additional property that bRG is dense whenever b is a right nonzero-divisor, then the complete ring of quotients of RG is a classical ring of quotients.     

RADICAL THEORY FOR SEMIRINGS

Abstract

In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P = um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals ?_∞(S) and ?^ε(S) and two similar ideals β_∞ (S) and β^ε(S) is widely solved. We prove ?_∞(S) ? ?^ε(S) = β^∞(S) = β^ε(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class M_ε(U) is always semisimple, a result which is not true for the special class M_∞(U).     

SUBDIRECT IRREDUCIBILITY AND RADICALS

Abstract

We prove lemmas in Andrunakievich s-varieties on the transitivity of the relation “is an ideal of” and concerning subdirectly irreducible factor rings. Applying these lemmas we show that a Plotkin radical introduced in [8] has the ADS-property and is ideal hereditary. These lemmas are applicable in proving a subdirect decomposition for rings having an ideal with 0 antisimple radical. For Jordan algebras and near-rings (they do not form Andrunakievich varieties) we can prove a similar subdirect decomposition concerning ideals with 0 Brown-McCoy radical.     

Graded Antisimple Radicals

We introduce the graded version of the antisimple radical S, the graded antisimple radical SG. When |G| < , we show that SG = Sref = SG, where Sref denotes the reflected antisimple radical and SG denotes the restricted antisimple radical.     

The conductor of one-dimensional gorenstein rings in their blowing-up

Let (A,M) be a local, one-dimensional, Cohen-Macaulay ring of multiplicity e=e(A)>1 and Hilbert function H(A). Let I=Ann_A (B/A) be the conductor of A in its blowing up B. Northcott and Matlis have proved that if the embedding dimension emdim A of A is 2 then I=M^e−1 [3; Corollary 13.8]. If emdim A>2 little is know about I. In [6] and [7] I is computed when the associated graded ring G(A) is reduced (in this case B in the integral closure of A). In this paper we compute I when A is Gorenstein. There are in general upper and lower bounds for I in terms of a power of M and we start discussing when these bounds are attained. In particular we show that in the extremal situation I=M^e−1 one has emdimA=2 (thus inverting the result of Northcott and Matlis). Then we consider the case of Gorenstein rings. We prove that if G(A) in Gorenstein then I=M^ϑ where ϑ=Min{n‖H(n)=e}. If more generally A is Gorenstein then I⊂M² or emdim A=e(A)=2. When A is the local ring of a curve at a singular point p we get, as a consequence of this last result a proof of the following conjecture of Catanese which has interesting geometric applications [1]: if the conductor J of A in its normalization is not contained in M² then p is a node.     

Gamma—rings and multiplications on abelian groups

If M and Γ are abelian groups, then M will be a Γ-ring iff there exists a group homomorphism f from Γ into the group of all multiplications of M, Mult(M), such that f(Γ) satisfies the Generalized Associativity Property on M. In this note we examine the following special cases of this result: (i) M is a Γ-ring satisfying the Nobusawa Condition, (ii) M is a cyclic group, (iii) M is a direct sum of cyclic groups and (iv) M is a Γ-ring that has unity elements.     

On multicyclic operators and the Vasjunin-Nikol'ski206-1206-1206-1discotheca

If A is a bounded linear multicyclic operator acting on a complex Banach spaceX, then thedisc of A is defined by: disc A = sup(R ∈ Cyc A) min{dimR′: R′ ? R, R′ ∈ Cyc A}, where Cyc A denotes the family of all finite dimensional subspacesR ofX such that X = (R+AR+A ² R+?)^?. It is shown that if the set {λ ∈ ?: dim ker (λ-A)^* ≥ n} has nonempty interior (in particular, if A is a Fredholm operator of index -n), then disc A ≥ n+1. This result affirmatively answers a question of V.I. Vasjunin and N.K. Nikol'skiï. In the case whenX is a Hilbert space, it is shown that the set of all operators A such that A is n-multicyclic, but disc A =∞, is dense in the set of all n-multicyclic operators. If M_λ = "multiplication by λ" acting on the disk algebra (and many other spaces of continuous and/or analytic functions), then M_λ is cyclic, but disc M_λ = ∞. However, the analogous result is false if the disk algebra is replaced by the algebra of functions analytic on the disk and smooth on the boundary, or algebras of Lipschitz functions. If T is a multicyclic unicellular operator, then T is cyclic and disc T=1.     

NOTES ON NILPOTENCY OF NIL SUBRINGS OF ENDOMORPHISM RINGS OF MODULES

Abstract

A family K of right R-modules is called a natural class if K is closed under submodules, direct sums, infective hulls, and isomorphic copies. The main result of this note is the following: Let K be a natural class on Mod-R and M ε K. If M satisfies a.c.c. (or d.c.c.) on the set of submodules {N ? M: M/N ε K}, then each nil subring of End(M_R ) is nilpotent.     

BOUNDS FOR INDICES OF NILPOTENCY OF FINITELY GENERATED NIL RINGS OF FINITE INDEX

Abstract

An improved bound is given for the index of nil-potency of a finitely generated nil ring of index n in terms of the index of nilpotency of the ideal generated by T^m where m = [n/2] and T is a m-subset of the set of generators. If m = 3 it is proved that T¹⁰ is contained in an ideal generated by twenty-seven cubes and this is applied to get bounds for the index of nilpotency of a finitely generated nil ring of index 6 or 7, bounds which are less than one hundredth of the bounds we obtained in a previous paper.     

ON CERTAIN NEW RADICALS ARISING FROM A GIVEN RADICAL

Abstract

If R is a ring and n is an integer weMaydefine a ring T_n (R) on the same underlying additive abelian group by using the formula a * b = nab to define a new multiplication. T_n , is a functor on the category of associative rings. If C is a class of rings then, for each n, the class C_n , is defined to consist of all rings R such that T_n (R) is in C. If C is a radical class then each class C_n , is also a radical class. We consider the properties of the radical class C which are inherited by C_n , and relationships between these classes C _n as n varies.     

*—BI—IDEALS IN RADICAL THEORY OF INVOLUTION RINGS

Abstract

If a radical class is closed under involution in every ring with involution then the radical theoretic conditions involving *-bi-ideals are equivalent to the corresponding conditions concerning bi-ideals without involution.     

Asymptotic Behavior of Solutions of Some Linear Difference Equations with Oscillatory Decreasing Coefficients

We investigate the asymptotic behavior of solutions of the system x ( n +1)=[ A + B ( n ) V ( n )+ R ( n )] x ( n ), n S n 0 , where A is an invertible m 2 m matrix with real eigenvalues, B ( n )= ~ j =1 r B j e i u j n , u j are real and u j p ~ (1+2 M ) for any M ] Z , B j are constant m 2 m matrices, the matrix V ( n ) satisfies V ( n ) M 0 as n M X , ~ n =0 X Á V ( n +1) m V ( n ) Á < X , ~ n =0 X Á V ( n ) Á 2 < X , and ~ n =0 X Á R ( n ) Á < X . If AV ( n )= V ( n ) A , then we show that the original system is asymptotically equivalent to a system x ( n +1)=[ A + B 0 V ( n )+ R 1 ( n )] x ( n ), where B 0 is a constant matrix and ~ n =0 X Á R 1 ( n ) Á < X . From this, it is possible to deduce the asymptotic behavior of solutions as n M X . We illustrate our method by investigating the asymptotic behavior of solutions of x 1 ( n +2) m 2(cos f 1 ) x 1 ( n +1)+ x 1 ( n )+ a sin n f n g x 2 ( n )=0 x 2 ( n +2) m 2(cos f 2 ) x 2 ( n +1)+ x 2 ( n )+ b sin n f n g x 1 ( n )=0 , where 0< f 1 , f 2 < ~ , 1/2< g h 1, f 1 p f 2 , and 0< f <2 ~ .