线性 J-Armendariz 环

摘要

如果对任意的 f (x )=a 0+a 1 x ,g (x )=b 0+b 1 x ∈R [x ],f (x )g (x )=0蕴含所有a ib j ∈J (R),则环 R 称为线性 J-Armendariz 环(简称 LJA 环)。其中：i ,j ∈{0,1}；J (R)是 R的 Jacobson 根。考虑 LJA 环的性质及与其他相关环类的关系,给出了2-primal 环的无限直积非2-primal 环的简单例子,并证明了 Koethe 猜想有肯定解当且仅当任意 NI 环的多项式环是 LJA 环。%A ring R is called LJA ring if f (x )g (x )=0 implies a ib j ∈J (R )for all f (x )=a 0 +a 1 x , g (x)=b 0 +b 1 x in R [x ]and i ,j = 0,1,where J (R )is the Jacobson radical of R.Considering the properties of LJA rings and the relationship between such rings and other related rings,the author gives simple examples of an infinite direct product of 2-primal rings not to be a 2-primal ring,and proves that Koethe’s conjecture has a positive solution if and only if a polynomial ring over any NI ring is an LJA ring.