Elementary Equivalence of I-injective Modules

Let R be a ring with 1 and consider the problem of assigning numerical invariants to right K-modules so that two such modules are elementarily equivalent if and only if they have the same invariants. This general problem has been solved by Baur [1], Garavaglia [8], and Monk [18]. The invariants they produce are however rather more model-theoretic than algebraic and are in general highly redundant. In this paper I consider the problem for a restricted class of modules but obtain invariants which have a direct algebraic significance. These invariants are of the Baur-Garavaglia-Monk type (see the comments at the end of the paper) but form a minimal set of such. The results here are considered from a fairly algebraic point of view. Some comments on a more model-theoretic approach are made at the end. Those with some knowledge of stability theory (in the model-theoretic sense) or who are acquainted with [9] might find it useful to refer to these comments and to bear them in mind while reading the paper. I have attempted to make this paper accessible both to algebraists and to modeltheorists. Thus for instance I have avoided the use of too powerful machinery from either area and have given some arguments with rather more detail than would normally be included. This applies in particular to arguments which are fairly routine but with which I would not expect all readers of this paper to be familiar. For definitions and background not included here I recommend [27] for the algebra and [25] or [4, §0] for the model theory. Throughout, R will be some fixed ring with 1 and 'module' will mean an object of the category JtR of right /^-modules. Associated with JtR is the usual first-order language Z£R: the variables of the language are to be thought of as varying over module elements; there is a constant symbol '0' for the zero element of a module; there is a symbol ' + ' to represent module addition; for each element r of R there is a function symbol (V written on the right of its argument without parentheses) with which to express multiplication of module elements by r. Thus, for example, '3u(/\"= 1 vrt = 0 A D ^ 0 ) ' where ri9...,rn e R, is a formula in this language (which says of any module which satisfies it that there is an element which kills each of rx,...,rn and which is non-zero—i.e. there is a non-zero element whose annihilator contains YJ=I «^)Note that quantification over elements of R is not possible in ££R. My main concern will be with elementary equivalence of injective modules. Two modules M,N are elementarily equivalent, M = N, if they satisfy precisely the same sentences of S£R. An embedding M ^ N is an elementary embedding, M<N, if the same is true when parameters from M are allowed to appear in the sentences. Write M =i N if M,N satisfy precisely the same existential sentences of S£R (those of the form 3vq)(v) where (p{v) contains no quantifiers, e.g. the sentence in the previous paragraph is such). The following result is easily proved using the above-mentioned