Armendariz半环的一些性质

Properties of Armendariz Semi-ring

在Armendariz环概念的基础上,定义了Armendariz半环,并将Anderson和Camillo提出的Armendariz环的一些性质推广到半环上;证明了在Armendariz半环R中,当n个一元多项式乘积为零时,其对应的系数乘积为零;其次,证明了R是Armendariz半环当且仅当R[x]是Armendariz半环;同时,给出了n个多元多项式乘积为零,其对应的系数乘积为零的等价条件;进一步证明了,当n≥2时,若R是约化半环,则R[x](x^n)为Armendariz半环。

Based on the concept of Armendariz rings,Armendariz semiring is defined,and some properties of Armendariz ring proposed by Anderson and Camillo are extended to semi-ring.It is proved that in the Armendariz-R,when the product of n unary polynomials is zero,their corresponding coefficient product is zero.Secondly,it is proved that R is an Armendariz semi-ring if and only if R[x]is an Armendariz semi-ring.At the same time,the equivalent condition is given that the product of n multivariate polynomials is zero and the product of their corresponding coefficients is zero.It is further proved that when n≥2,if R is a reduced semi-ring,then R[x](x^n)is an Armendariz semi-ring.