Structure of rings satisfying certain polynomial identities

A well-known theorem of Jacobson [2] asserts that if $R$ is an associative ring with the property that, for all $x$ in $R$ , there exists an integer $m(x)>1$ such that $x^{m(x)}=x$ , then $R$ is isomorphic to a subdirect sum of fields.Our present object is to extend Jacobson's Theorem by determining the structure of a certain class of associative rings satisfying polynomial identities involv- ing $n$ elements $x_{1}$ , $\cdot$ .. , $x_{n}$ of $R$ .In order to be able to state this generaliza- tion, we first define a word $w(x_{1}$ , $\cdot$ .. , $x_{n})$ in $x_{1}$ , $\cdot$ .. , $x_{n}$ to be a product in which each factor is $x_{i}$ for some $i=1$ , $\cdot$ .. , $n$ .A polynomial $f(x_{1}$ , $\cdot$ .. , $x_{n})$ is, then, an expression of the form $c_{1}w_{1}(x_{1}$ , $\cdot$ .. , $x_{n})+\cdots+c_{m}w_{m}(x_{1}$ , $\cdot$ .. , $x_{n})$ , where the $c_{i}$ are integers.The degree of $x_{i}$ in the word $w(x_{1}, \cdots , x_{n})$ is the number of times $x_{i}$ appears as a factor in $w(x_{1}$ , $\cdot$ .. , $x_{n})$ .Suppose that $f(x_{1}$ , $\cdot$ ., , $x_{n})=c_{1}w_{1}(x_{1}$ , , $x_{n})+$ $+c_{m}w_{m}(x_{1}, x_{n})$ is a polynomial in $x_{1},$ $x_{n}$ .The degree of $x_{i}$ in $f(x_{1}$ , $\cdot$ .. , $x_{n})$ is the smallest value among the following: degree of $x_{i}$ in $w_{1}(x_{1}$ , ... , $x_{n}$), $\cdots$ degree of $x_{i}$ in $w_{m}(x_{1}, \cdots , x_{n})$ .The following theorem is proved: