【摘要】 本文给出了s-单式环的几个交换性定理及一般环的一个交换性定理。更多还原

【Abstract】 In this paper we prove the following Commutativity theorems on rings.Theorem 1 Let s-unital ring R satisfies the identities for all x,y∈R where n,s,t>1 are fixed integers. If R is (s,t)-torsion free, then R is commutative.Theorem 2 Let R be an s-unital ring. If R satisfies one of the following conditionsP) for any x,y∈R there exist m,s,t,m’,s’,t’>1, (t,t’)=1. such thatQ) for any x,y∈R there exist m,s,t,m’,s’,t’>1, (s+1,s’+1)=1 such that then R is commutative.Theorem 3 Let R be an s-unital semi-prime ring. If R satisfies one of the following conditionsi) for any x, y∈R there exist integers n,s,t>1 such that ii) for any x,y∈R there exist integers n,s,t>l such that then R is commutative. Let A be a subset of R. We study the following conditions of R.(I-A) For x∈R there exists a polynomial f(t) with integral coefficients such that x-x~2f(x)∈A.(Ⅱ-A) If x,y∈R, x-y∈A then either x~m=y~m, x~n=y~n, m=m(x, y), n=n(x,y), (m,n)=1 or x,y∈C_R(A).(Ⅲ-A) If x,y∈R, x-y∈A then either [x, (x+y)~2]=[x,y~2] r x,y∈C_R(A).We haveTheorem 4 Let A be a nil commutative subset of R such that if a∈A then -a∈A, then1) R is commutative if R satisfies conditons (I-A) and (Ⅱ-A).2) R is commutative if R satisfies conditions (Ⅰ-A) and (Ⅲ-A).更多还原