Product of ring varieties and attainability

Abstract

The class of all rings that are Everett extensions of a ring in a variety U \mathfrak {U} by a ring in a variety B \mathfrak {B} is a variety U ⋅ B \mathfrak {U} \cdot \mathfrak {B} . With respect to this operation the set of all ring varieties is a partially ordered groupoid (under inclusion), that is not associative. A variety is idempotent iff it is the variety of all rings, or generated by a finite number of finite fields. No families of polynomial identities other than those equivalent to x = x x = x or x = y x = y are attainable on the class of all rings or on the class of all commutative rings.