ON THE COMMUTAIIVITY AND CHARACTERISTIC OF RINGS

This paper is devoted to discussing the connections between the commutativi-ty and characteristic of a non-zero-divisor ring with identity. The paper mainly provs the following conclusion:Theorem 1 If R is a non-zero-divisor ring with identity, moreover,then (CharR-l)|(p-l)Theorem 2 Let R be a non-zero-divisor ring with e as identity, |R|≥p,moreover, for any a∈R, (a + e)p=ap + e then Char R=pTheorem 3 Provided that R is a non-zero-divisor ring with unit, there exists a prime p1, such that p≠CharR, moreover, for any a∈R, (a+e)p=ap+e then R is a finite field.Theorem 4 If 1) R is an identity preserving ring with the characteristic 0 and non-zero-divisor, 2 ) for any x,y in R, there exist the integers a1,a2,a3,b1,b2, b3 such that:a1xy2+ a2yxy+a3x2y+b1xyx + b2yx2+b3y2x=0(6)then we reach that:If R is commutative, then(a1+2b3)(2a1 + a2)(b2+2a3)(2b1+b2)≠0 (7)Contrariwise, if there exists a non-zero-divisor among the left-hand divisors in the equation (7) , then R is a cpmmutative ring.