A Result of Levitzki Tyte with Annihilator Conditions on Multilinear Polynomials

摘要

Let R be a prime K-algebra without nonzero nil right ideals and f(X_1,··· , X_t) a multilinear polynomial over K, where K is a commutative ring with unity. Suppose that (a f(x_1,··· , x_t))~nb = 0 for some nonzero elements a, b ∈ R and for all x_1,··· , x_t in some nonzero ideal of R, where n = n(x_1,··· ,x_t) is a positive integer depending on x_1,··· ,x_t. Then f(X_1,··· ,X_t) is central-valued on R. As a consequence of this theorem a parallel result is also obtained for semiprime K-algebras.