It is known that the existence
of a nonzero commutative one-sided ideal in a prime ring implies that the whole ring
is commutative. Since rings satisfying a polynomial identity are natural
generalizations of commutative rings the question arises as to what extent the above
mentioned result can be extended to include these generalizations. That
is, if R is a prime ring and I a nonzero one-sided ideal which satisfies a
polynomial identity does R satisfy a polynomial identity? This paper initiates an
investigation of this problem. A counter example, given later, will show that the
answer to the above question may be negative, even when R is a simple
primitive ring with nonzero socle. The main theorem of this paper is Theorem
3 which states: Let R be a prime ring having a nonzero right ideal which
satisfies a polynomial identity. Then, a necessary and sufficient condition
that R satisfy a polynomial identity is that R have zero right singular ideal
and R, the right quotient ring of R, have at most finitely many orthogonal
idempotents.
