THE VARIETY OF RINGS WITH INVOLUTION SATISFYING x7≈ x

We give a complete description of the lattice of varieties of rings with involution satisfying x7≈ x by identity bases. There are 90 such varieties. If we substitute in each ring of such a variety the operations by term operations of the same arity we obtain a so-called class of derived rings. We discuss the case when the class of derived rings belongs to the original variety. In particular, we describe the class of derived rings for the variety of rings generated by the two-element Galois-field.

Injective and Weakly Injective Rings

Let V be a variety of rings and let A ∊ V. The ring A is injective in V if every trianglewith C ∊ V, m a monomorphism and f a homomorphism has a commutative completion as indicated. A ring which is injective in some variety (equivalently, injective in the variety it generates) is called injective. When only triangles with f surjective are considered we obtain the notion of weak injectivity. Directly indecomposable injective and weakly injective rings are classified.

Two Undecidability Results using Modified Boolean Powers

In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction.