This paper gives variants of results from classical algebraic geometry and commutative algebra for quadratic algebras with conjugation. Quadratic algebras are essentially two-dimensional algebras with identity over commutative rings with identity, on which a natural operation of conjugation may be defined. We define the ring of conjugate polynomials over a quadratic algebra, and define c-varieties. In certain cases a close correspondence between standard varieties and c-varieties is demonstrated, and we establish a correspondence between conjugate and standard polynomials, which leads to variants of the Hilbert Nullstellensatz if the commutativering is an algebraically closed field. These results may be applied to automated Euclidean geometry theorem proving.

The Brown-McCoy radical is the upper radical defined by the class of simple rings with identities. For associative or alternative rings the Brown-McCoy radical is hereditary, and its semi-simple class consists of all subdirect products of simple rings with identities. In this paper we present some classes of simple non-associative algebras whose upper radicals behave similarly. Classifications are then obtained of ‘most’ semi-simple radical classes of (γ, δ) and right alternative rings.

An example is constructed of a locally finite variety of non-associative algebras which satisfies the maximal condition on subvarieties but not the minimal condition. Based on this, counterexamples to various conjectures concerning varieties generated by finite algebras are constructed. The possibility of finding a locally finite variety of algebras which satisfies the minimal condition on subvarieties but not the maximal is also investigated.