Vol. 24, No. 3, 1968

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Prime rings with a one-sided ideal satisfying a polynomial identity

Lawrence Peter Belluce and Surender Kumar Jain

Vol. 24 (1968), No. 3, 421–424

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.

Mathematical Subject Classification
Primary: 16.49
Received: 9 October 1964
Published: 1 March 1968
Lawrence Peter Belluce
Surender Kumar Jain