线性J-Armendariz环

如果对任意的f(x)=a₀+a₁x, g(x)=b₀+b₁x∈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.

Linear J Armendariz Rings

A ring R is called LJA ring if f(x)g(x)=0 implies a_ib_j∈J(R) for all f(x)=a₀+a₁x, g(x)=b₀+b₁x 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.