A prime model O of some complete theory T is a model which can be elementarily imbedded into any model of T (cf. Vaught [7, Introduction]). We are going to replace the assumption that T is complete and that the maps between the models of T are elementary imbeddings (elementary extensions) by more general conditions. T will always be a first order theory with identity and may have function symbols. The language L(T) of T will be denumerable. The maps between models will be so called F-maps, i.e. maps which preserve a certain set F of formulas of L(T) (cf. I.1, 2). Roughly speaking a generalized prime model of T is a denumerable model O which permits an F-map O→M into any model M of T. Furthermore O has to be “generated” by formulas which belong to a certain subset G of F.

It was shown by Taiclin [6], and independently announced by Tarski [7], that the elementary theory of commutative cancellation semigroups is hereditarily undecidable. In his proof Tarski exhibited a subsemigroup of 〈N, ·〉, the natural numbers with multiplication, whose theory is both hereditarily and essentially undecidable. (The details of his construction were published by V. H. Dyson [1].) In connection with these results, Tarski suggested to the author that it would be of interest to solve the decision problem for the theory K which consists of all elementary sentences which are true in every subalgebra (i.e. every subsemigroup) of 〈N, +〉. The object of this note is to prove that the theory K is hereditarily undecidable.

Lindenbaum's construction of a matrix for a propositional calculus, in which the wffs themselves are taken as elements and the theorems as the designated elements, immediately establishes two general results: that every prepositional calculus is many-valued, and that every many-valued propositional calculus is also ℵ0-valued. These results are however concerned exclusively with theoremhood, the inferential structure of the calculus being relevant only incidentally, in that it may serve to determine the set of theorems. We therefore ask what happens when deducibility is taken into consideration on a par with theoremhood. The answer is that in general the Lindenbaum construction is no longer adequate and both results fail.

This paper is concerned with questions of the following kind: let L be a language of the form Lαω and let be a class of modules over a fixed ring or a class of rings; is it possible to define in L? We will be mainly interested in the cases where L is Lωω or L∞ω and is a familiar class in homologic algebra or ring theory.

In Part I we characterize the rings Λ such that the class of free (respectively projective, respectively flat) left Λ-modules is elementary. In [12] we solved the corresponding problems for injective modules; here we show that the class of injective Λ-modules is definable in L∞ω if and only if it is elementary. Moreover we identify the right noetherian rings Λ such that the class of projective (respectively free) left Λ-modules is definable in L∞ω.

Throughout this paper, the letters α, β, κ, λ will denote infinite cardinals. Other Greek letters range over the ordinals.

The following definition is taken from Chang [1].

This paper resolves 3 problems left open by R. M. Robinson in [3].

We recall that the set of primitive recursive functions is the closure under (i) substitution (or “composition”), and (ii) recursion, of the set P consisting of the zero, successor and projection functions (see any textbook, for instance p. 120 of [2]).