Rings satisfying monomial constraints

Abstract:

The following theorem is proved: Suppose $R$ is an associative ring and suppose $J$ is the Jacobson radical of $R$. Suppose that for all ${x_1}, \cdots ,{x_n}$ in $R$, there exists a word ${w_{{x_1}}}, \cdots ,{x_n}({x_1}, \cdots ,{x_n})$, depending on ${x_1}, \cdots ,{x_n}$, in which at least one ${x_i}$ ($i$ varies) is missing, and such that ${x_1} \cdots {x_n} = {w_{{x_1}, \cdots ,{x_n}}}({x_1}, \cdots ,{x_n})$. Then $J$ is a nil ring of bounded index and $R/J$ is finite. It is further proved that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent.