Abstract:

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.

Key words: Armendariz ring, linear Armendariz ring, linear J-Armendariz ring