ON ANTISIMPLE GAMMA RINGS

ABSTRACT

In this note, we define the antisimple radical, A(M), of a Γ-ring M. A(M) is shown to be a special radical, and two characterizations of antisimple rings due to Szész are extended to Γ-rings. If R is the right operator ring of M, then A(R)* = A(M), where A(R) is the antisimple radical of R. If m,n are positive integers, then A(M_mn) = (A(M))_mn, where M_mn denotes the group m x n matrices over M, considered as a Γ_nm -ring with the operations of matrix addition and multiplication.     

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.     

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.     

Nil-Clean Rings with Involution

A *-ring R is called a nil *-clean ring if every element of R is a sum of a projection and a nilpotent.Nil *-clean rings are the *-version of nil-clean rings introduced by Diesl.This paper is about the nil *-clean property of rings with emphasis on matrix rings.We show that a *-ring R is nil *-clean if and only if J(R) is nil and R/J(R) is nil*-clean.For a 2-primal *-ring R,with the induced involution given by (aij)* =(a*ij)T,the nil *-clean property of Mn(R) is completely reduced to that of Mn(Z2).Consequently,Mn(R) is not a nil *-clean ring for n =3,4,and M2(R) is a nil *-clean ring if and only if J(R) is nil,R/J(R) is a Boolean ring and a*-a ∈ J(R) for all a ∈ R.     

ON ORDER RINGS OF SEMI-PRIMARY RINGS

The main results of this paper are stated as follows.Let R be an orderring in thesemi-primary ring Q.Suppose that R satisfies the maximal condition for nil right ideals ofR,Then we have(i)if an ideal I of R has a finite length as right R-module,then I alsohas a finite length as left R-module;(ii)denote by A(R)the Artinian radical of R,andN the nil radical of R,then A(R)+N/N=A(R/N),if R satisfies the commutative condi-tion on the zero product of prime ideals of B.     

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.     

A generalization of semi prime indeals in γ-rings

The purpose of this paper is to generalize the concept of semi prime ideals in Γ-rings. We use a general definition of a regularity F for Γ-rings to define and F- prime ideal. Relationships between F-semi prime ideals of a Γ-ring M and F-semi prime ideals of the operator rings R and L are discussed. D-regularity, f-regularity and λ-regularity for Γ-rings are introduced and studied against the background of the concept F-semi prime ideal. Finally, D-, λ- and f-regular Γ-rings are characterized.     

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.     

PRME AND k-PRIME IDEALS IN Ω-GROUPS

Abstract

In this paper we define two concepts of prime ideals for Ω-groups. The first generalizes the definitions of prime ideal in rings, nearrings, Γ-rings, associative algebras and Lie algebras. The second generalizes a concept defined for groups by ??ukin ([21]). We show that both lead to radicals in the sense of Hoehnke ([10]). Furthermore in the case of rings, Γ-rings, abelian zero-symmetric nearrings and cubic rings these two definitions coincide, thus obtaining a new characterization for the prime ideal. Zero-symmetric Ω-groups are defined analogously to the nearring case and a new characterization in term of ideals is given.     

分次广义Γ-环的Brown-McCoy根

研究了分次广义Γ-环的弱强分次B row n-M cC oy根与拟弱强分次B row n-M cC oy根,并从不同角度刻划了分次广义Γ-环的弱强分次B row n-M cC oy根.证明了任何一个分次广义Γ-环都有弱强分次B row n-M cC oy根和拟弱强分次B row n-M cC oy根,而且弱强分次B row n-M cC oy根小于等于拟弱强分次B row n-M c-C oy根.     

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.     

A NOTE ON LEFT STROUGLY NIL AND LEFT STROUGLY NILPOTENT

Abstract

In [2] van der Walt called a left ideal L of a ring A, left strongly nil, if given 1 ε L and k ε K, K a left ideal. there is an n such that (1+k)ⁿ ε K. L is called left strongly nilpotent if for any left ideal K there exists an m such that (L+K)^m ? K. In this paper we will prove that if A is a left artinian ring (not necessarily with unity) then every left strongly nil left ideal is left strongly nilpotent. This result is a generalization of the main theorem of [2].     

The Zero Divisor Graph of 2 × 2 Matrices Over a Field

A zero divisor graph, Γ(R), is formed from a ring R by having each element of Z(R) \ {0} to be a vertex in the graph and having two vertices u and v adjacent if the corresponding elements from the ring are nonequal and have product equal to zero. In this paper, the structure of the zero-divisor graph of 2 × 2 matrices over a field, Γ(M2(F)), are completely determined.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

NILPOTENCY,SOLVABILITY AND RADICALS IN CATEGORIES

Abstract

Nilpotent and solvable ideals are defined and investigated in categories. The relation between the prime radical and the sum of the solvable ideals (which is also a radical) is discussed in categories. For example: If an object satisfies the maximal condition for ideals, then the prime radical is equal to the sum of the solvable ideals. Certain generalizations of theorems in rings, groups, Lie algebras, etc. are also proven, for example: An ideal α: I → A is semiprime if and only if A/I contains no non-zero nilpotent ideals.     

TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

Abstract

Transcendental and algebraic elements over commutative rings are defined. Rings with zero nil radical are considered. For a transcendental over R, necessary and sufficient conditions are derived for elements of R[α] to be algebraic or transcendental over R. For R a ring with identity and a finite number of minimal prime ideals, necessary and sufficient conditions are given for any element in a unitary overring of R to be algebraic or transcendental over R. It is proved that if α is algebraic Over R, so is every element of R[α]. It is show that if R is Noetherian, β is algebraic over R[α] and α is algebraic over R, then, under certain conditions, β is algebraic over R. If R has a finite number of minimal prime ideals, P₁,…,P_k, which are pairwise comaximal, then if t is transcendental over R, R[t] can be obtained by adjoining k algebraic elements a_i over R to R whose defining polynomials are in P_i [x], and conversely, if such elements are adjoined to R, they generate an element transcendental over R.     

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.