强诣零Armendariz环  

Stronglynil-Armendarz Rings

作　　者：崔书英[1] 

机构地区：[1]山东工商学院数学与信息科学学院,山东烟台264005

出　　处：《邯郸学院学报》2010年第3期30-34,共5页Journal of Handan University

基　　金：山东省自然科学基金资助课题(课题编号:Y2008A04)

摘　　要：结合环R称为强诣零Armendariz的如果对于R[x]中任意两个多项式f(x),g(x)当f(x)g(x)∈Nil*(R)[x]时,有ab∈Nil*(R),这里a,b分别是f(x),g(x)的任何系数,而N*(R)为R的素根。证明了强诣零Armendariz环R的素根与上诣零根一致;强诣零Armendariz环是诣零Armendariz环;证明了R是强诣零Armendariz环当且仅当R的每个子环是强诣零Armendariz环,当且仅当R的多项式环R[x]是强诣零Armendariz环,当且仅当R的上三角矩阵环Tn(R)是强诣零Armendariz环;R是强诣零Armendariz环当且仅当R/Nil*(R)是Armendariz环。并推广了弱Armendariz环的两个结果。Rings are associative with identity and subrings may not have identity.A ring R is called strongly nil-Armendariz if for f（x）,g（x）∈ R[x]whenever f（x）g（x）∈ Nil＊（R）[x]then ab ∈ Nil＊（R）where a,bare any coefficients of f（x）,g（x）and Nil ＊（R）is the prime radical of R.We prove that Nil ＊（R）= Nil＊（R）for a strongly nil-Armendariz ring R,and that a strongly nil-Armendariz ring is a nil-Armendariz ring.And it is proved that a ring R is strongly nil-Armendariz if and only if each subring of R is strongly nil-Armendariz,if and only if R[ x]is strongly nil-Armendariz,if and only if Tn（R） is strongly nil-Armendariz,R is strongly nil-Armendariz if and only if R / Nil＊（R）is Armendariz.Moreover we generalize two known results on weak Armendariz rings.

关 键 词：ARMENDARIZ环 诣零Armendariz环 强诣零Armendariz环 弱ARMENDARIZ环 

分 类 号：O153[理学—数学]