Commutativity for a Certain Class of Rings
نویسنده
چکیده
We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr[xs, y] = ±[x, yt]xn, xr[xs, y] = ±xn[x, yt], xr[xs, y] = ±[x, yt]ym, and xr[xs, y] = ±ym[x, yt], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we improve the result of [6, Theorem 3] and reprove several recent results. Throughout the paper R will represent an associative ring (with or without unity 1). Let C(R) denote the commutator ideal of R, Z(R) the center of R, and H the heart of R. By (GF (q))2 we mean the ring of 2 × 2 matrices over the Galois field GF (q) with q elements. Set e11 = ( 1 0 0 0 )
منابع مشابه
A COMMUTATIVITY CONDITION FOR RINGS
In this paper, we use the structure theory to prove an analog to a well-known theorem of Herstein as follows: Let R be a ring with center C such that for all x,y ? R either [x,y]= 0 or x-x [x,y]? C for some non negative integer n= n(x,y) dependingon x and y. Then R is commutative.
متن کاملOn centralizers of prime rings with involution
Let $R$ be a ring with involution $*$. An additive mapping $T:Rto R$ is called a left(respectively right) centralizer if $T(xy)=T(x)y$ (respectively $T(xy)=xT(y)$) for all $x,yin R$. The purpose of this paper is to examine the commutativity of prime rings with involution satisfying certain identities involving left centralizers.
متن کاملSome commutativity theorems for $*$-prime rings with $(sigma,tau)$-derivation
Let $R$ be a $*$-prime ring with center $Z(R)$, $d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated automorphisms $sigma$ and $tau$ of $R$, such that $sigma$, $tau$ and $d$ commute with $'*'$. Suppose that $U$ is an ideal of $R$ such that $U^*=U$, and $C_{sigma,tau}={cin R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper, it is shown that if charac...
متن کاملCommutativity degree of $mathbb{Z}_p$≀$mathbb{Z}_{p^n}
For a nite group G the commutativity degree denote by d(G) and dend:$$d(G) =frac{|{(x; y)|x, yin G,xy = yx}|}{|G|^2}.$$ In [2] authors found commutativity degree for some groups,in this paper we nd commutativity degree for a class of groups that have high nilpontencies.
متن کاملOn co-Noetherian dimension of rings
We define and studyco-Noetherian dimension of rings for which the injective envelopeof simple modules have finite Krull-dimension. This is a Moritainvariant dimension that measures how far the ring is from beingco-Noetherian. The co-Noetherian dimension of certain rings,including commutative rings, are determined. It is shown that the class ${mathcal W}_n$ of rings with co-Noetherian dimension...
متن کاملHomomorphism Weak amenability of certain Banach algebras
In this paper we introduce the notion of $varphi$-commutativity for a Banach algebra $A$, where $varphi$ is a continuous homomorphism on $A$ and study the concept of $varphi$-weak amenability for $varphi$-commutative Banach algebras. We give an example to show that the class of $varphi$-weakly amenable Banach algebras is larger than that of weakly amenable commutative Banach algebras. We charac...
متن کامل