The Tensor Products of Nilrings with Bounded Index of Nilpotence

R is a ring, for any nilpotent element r∈R if there exists a fixed integer nsuch that r~n=0, then R is said to be with bounded index of nilpotence, theleast of such integer n is called the index of R, denoted by i(R). If R is a nil ring with bounded index i(R)=n, R′ is a commutative ring,A·A·Klein〔1〕 has discussed the property of bounded index of nilpotence of RR′. In this paper we shall discuss the properties of bounhed index of nil-potence of RR′ when R′ is not commutative.     

TWO COMMUTATIVITY RESULTS FOR SEMI-PRIME RINGS

1.Introduction.In 1980.V.Gupta [2] provedTheorem A Let R be a semi-prime ring with unit satisfying(i)[x~n,y]-[x,y~n]∈Z(R) (ii)[x~(n+1),y]-[x,y~(n+1)]∈Z(R)for all x,y∈R and a fixed integer n>1,then R is commutative.This theoremimproved a theorem which was established by Harmanci [4] in 1977 that if aring R with unit satisfies the identity(i)[x~n,y]=[x,y~n] (ii)[x~(n+1),y]=[x,y~(n+1)] for all x,y∈Rand a fixed integer n>1,then R is commutative.Later,in 1982,Guo Yuan     

具有对称自同态的环及其扩张

Let R be a ring with an endomorphism α and an α-derivation δ. We introduce the notions of symmetric α-rings and weak symmetric α-rings which are generalizations of symmetric rings and weak symmetric rings, respectively, discuss the relations between symmetricα-rings and related rings and investigate their extensions. We prove that if R is a reduced ring and α(1) = 1, then R is a symmetric α-ring if and only if R[x]/(x n) is a symmetric ˉα-ring for any positive integer n. Moreover, it is proven that if R is a right Ore ring, α an automorphism of R and Q(R) the classical right quotient ring of R, then R is a symmetric α-ring if and only if Q(R) is a symmetric ˉα-ring. Among others we also show that if a ring R is weakly 2-primal and(α, δ)-compatible, then R is a weak symmetric α-ring if and only if the Ore extension R[x; α, δ] of R is a weak symmetric ˉα-ring.     

关于斜McCoy环

For a ring endomorphism α,we introduce α-skew McCoy rings which are generalizations of α-rigid rings and McCoy rings,and investigate their properties.We show that if α t = I R for some positive integer t and R is an α-skew McCoy ring,then the skew polynomial ring R[x;α] is α-skew McCoy.We also prove that if α（1） = 1 and R is α-rigid,then R[x;α]/ x 2 is αˉ-skew McCoy.     

正则环和GP-内射环

A ring R is called left Gp-injective if for any a∈R, there exists a positive integer n such that any left R-homomorphism of Raⁿ into R extends to one of R into R. In this paper, we prove that the centre of semiprime (left nonsingular) left GP- injective ring is regular ring, and improve some propositions in [3].     

BOUNDEDNESS OF INFINITE DELAY DIFFERENCE SYSTEMS IN TERMS OF TWO MEASURES

1 PreliminariesLet R (R--), Z (Z--) denote the sets of non-negative (non-positive) realnumbers and nonnegative (nonpositive) integers, respectively, X= {of: { --r,'' 1--2, --1, 0} - Rk}, where r is a non-negative integer or r = oo. DenoteF == {h: Z X Rk - R , h(n, x) is continuous in x, and inf{h(n, x)} = 0},K = {a E C(R ,R ) t a(u) is strictly increasing in u and a(0) = 0},n LQ = {ry E C(R , R ): there are constants a, L 2 1 such that Z n(s) < a,s=n 1for all n E Z }, and in this …     

具有对合运算的$n$-clean环

A *-ring is called *-clean if every element of the ring can be written as the sum of a projection and a unit. For an integer n ≥ 1, we call a *-ring R n-*-clean if for any a ∈ R,a = p + u1 + ··· + unwhere p is a projection and ui are units for all i. Basic properties of n-*-clean rings are considered, and a number of illustrative examples of 2-*-clean rings which are not *-clean are provided. In addition, extension properties of n-*-clean rings are discussed.     

Notes on Automorphisms of Prime Rings

Let R be a prime ring, L a noncentral Lie ideal and σ a nontrivial automorphism of R such that usσ(u)ut= 0 for all u ∈ L, where s, t are fixed non-negative integers. If either char R s + t or char R = 0, then R satisfies s4, the standard identity in four variables. We also examine the identity(σ([x, y])-[x, y])n=0 for all x, y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. If either char R n or char R = 0, then R is commutative.     

On the differential uniformities of functions over finite fields

In this paper, the possible value of the differential uniformity of a function over finite fields is discussed. It is proved that, the differential uniformity of a function over Fq can be any even integer between 2 and q when q is even; and it can be any integer between 1 and q except q-1 when q is odd. Moreover, for any possible differential uniformity t, an explicit construction of a differentially t-uniform function is given.     

On Ding homological dimensions

In this paper, we investigate Ding projective dimensions and Ding injective dimensions of modules and ringsLet R be a ring with r DP D(R) = n ∞, and let W1 = {M|fd(M) ∞}We prove that(DP, W1) is a complete hereditary cotorsion pair such that a module M belongs to DP ∩ W1 if and only if M is projective, moreover,W1 = {M|pd(M) ∞} = {M|fd(M) ≤ n} = {M|pd(M) ≤ n}Then we introduce and investigate Ding derived functor Dexti(-,-), and use it to characterize global Ding dimensionWe show that if R is a Ding-Chen ring, or if R is a ring with r DP D(R) ∞ and r DI D(R) ∞,then r DP D(R) ≤ n if and only if r DI D(R) ≤ n if and only if Dextn+i(M, N) = 0 for all modules M and N and all integer i ≥ 1.