On the Variety Determined by Symmetric Quadratic Algebras

Abstract

We consider some polynomial identities of degree ≤5 which are satisfied by all symmetric quadratic algebras. We call rings satisfying these identities generalized quadratic rings, or GQ-rings. We show that when the ring is not flexible, these identities are enough to make the ring quadratic over its center. Therefore, simple nonflexible GQ-rings are symmetric quadratic algebras over their center, which is a field. For prime GQ-rings, the center has no nonzero zero divisors. Prime GQ-rings, which are not flexible, are subrings of the quadratic algebra formed by extending the center to its field of quotients. Flexible GQ-rings are noncommutative Jordan rings which satisfy a quadratic condition over their commutative center. We show that any semi-prime GQ-ring is a subdirect sum of a noncommutative Jordan ring (which satisfies a quadratic condition over its commutative center) and a nonflexible ring (which is symmetric quadratic over its center).