EXPLICIT DESCRIPTIONS OF THE INDECOMPOSABLE INJECTIVE MODULES OVER JATEGAONKAR'S RINGS

Abstract

ABSTRACT In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of the indecomposables is preserved by the localizations. An indecomposable injective is the union of an ascending chain (of order type at most ) of extensions by finite dimensional vector spaces. In the case of non-commutative (one- or two-sided) noetherian rings, the connection between the structure of injective modules and the behaviour–-even the possibility–-of localization remains (see, e.g., Jategaonkar[4] for an extended argument as to why this is so), but very little is understood about how the indecomposable injectives are put together in general. Indeed, the book of Jategaonkar just cited provides ample evidence that indecomposable injectives over one-sided noetherian rings can have wildly varying kinds of structure, and that some examples may be, in some sense, inaccessible to our understanding. In remarking about the difficulties associated with one-sided noetherian rings, Jategaonkar[[4], p. 91] says “Our decision to work with Noetherian rings rather than right Noetherian ones is often dictated by the exigencies of the situation under consideration. We note though that, after Jategaonkar (69) [that is[3], of the current bibliography], an attempt to stay with right Noetherian rings at all costs is generally regarded as futile.” The family of examples developed in the paper cited are important and fascinating: they simultaneously provide counter-examples to a handful of different conjectures of the time, and they illustrate how strongly the left ideal structure of a ring can be disconnected from the right ideal structure of the ring. Nonetheless, they do not provide examples of indecomposable injectives that are difficult to understand or impossible to describe. It is the purpose of this paper not only to describe the indecomposable injectives over these badly behaved rings, but to convince the reader that the structure of the indecomposables is easily and naturally deduced from elementary facts about injective modules combined with the description of the rings themselves. In actual fact, this latter task would take up far too much paper for the family of rings in[3]. Instead, I start with a much simplified version of these rings, the ring treated in Example 3.3.8 of Jategaonkar's book.[4] This ring is just a homomorphic image of the first member of the more general family. After describing the ring, in Sec. 2 I take the reader step-by-step through a natural process (complete with one intentional mistaken initial guess!) that leads to explicit descriptions of the two indecomposable injectives over the ring . The description is sufficiently explicit to make it routine to check, after the fact, that the structures that we have deduced actually are the indecomposable injectives for which we were looking. This is followed in Sec. 3 by showing how this explicit description can be used to solve certain kinds of problems related to the indecomposables. The process of showing how to deduce from elementary facts what the indecomposable injectives look like in the general version of the examples is not any more instructive than the shorter exposition presented for the simple version of the example. As a result, in the final section I content myself with describing the examples and verifying their correctness. This is in fact in itself a fairly lengthy task. I conclude the section by applying the explicit descriptions to analyze some of the internal structure of these indecomposable injectives. The idea that such constructions might be possible was inspired in part by my work on some explicit descriptions of complicated indecomposable injective modules over certain commutative noetherian rings[6] and by example 9.3.7 in Jategaonkar's book,[4] where the same sort of construction is used to build an explicit description in a non-commutative noetherian case. The opportunity to read an early version of Musson's paper[11] also helped. The approach taken here to injective modules is to view them as structures in which we can solve systems of linear equations. For more details on this approach and its history, and in particular on the equivalency with the usual definition, see my earlier paper.[6] Here I just repeat enough to ensure readability of this paper. A linear equation (in variables ) (over a right -module ) is just an -linear combination of the variables in set equal to some constant from . A system of linear equations in possibly infinitely many variables over is a set (again possibly infinite) of linear equations over , each in finitely many variables from . A solution to such a system in some is just an assignment of values in to each of the variables of that makes each of the equations true in . Note that of course a finite system of equations can be presented in matrix form as , where is a matrix over and is a tuple in the right -module . Such a finite system of equations is consistent if for every matrix over (of the right shape) such that , we also have . An infinite system of linear equations is consistent if every finite subset of it is consistent. It is straightforward to see that a system of equations over is consistent if and only if it has a solution in some extension of . A right -module is injective if every consistent system of linear equations over (possibly infinite, with infinitely many variables) has a solution in itself. Baer's criterion for injectivity in this formulation reads: is injective if every (possibly infinite) system of linear equations over in one variable has a solution in itself. I remind the reader that the injective envelope of the right-module is characterized variously as a maximal essential extension of , as a minimal injective extension of , or as an injective essential extension of . (In the language of linear equations, a module is an essential extension of its submodule if and only if every non-zero element of satisfies a non-trivial linear equation over .) The injective envelope of always exists. An injective module is (direct sum) indecomposable if and only if it is the injective envelope of for some meet-irreducible right ideal . Much of the work here has been motivated by my work in the model theory of modules. The main body of the material presented here requires no knowledge of mathematical logic or the model theory of modules, but some of the applications of these results are to problems in the model theory of modules (Sec. 3.1, 3.3). Notation. The symbol is used to denote the submodule ideal, etc generated by “”. Scalar multiplications are indicated by a large dot: . Ordered pairs are enclosed in angle brackets: .