Canadian Journal of Mathematics, Vol. 24, No. 4 (1972), 573-579

Received June 17, 1971 and in revised form, December 22, 1971. This paper will form part of the author's Ph.D. thesis.

Introduction

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of "algebraic" extensions of general algebraic systems has been studied by Shoda.)

In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules

f : I → M (I a left ideal of R)

which will be call an *ideal homomorphism* into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism to a homomorphism f* from R into M, or into an essential extension of M. Since such an extension f* of f is completely determined by f*(1), any element of the form x = f*(1) is an essential extension E of M will be called a *root of* f (in E).

The key to the analogy between algebraic closure and injectivity is given by "Baer's criterion for injectivity" which states, in the terminology above: Given _{R}M, if every ideal homomorphism into M has a root in M, then M is injective.

To continue the analogy, we define the *splitting module*, over M, of a set of ideal homomorphisms f_{j} : I_{j} → M to be the submodule S of H(M) generated by M and all the roots in H(M) of the given homomorphism; that is

S = M + ∑ Rx (x = f_{j}*(1))

where the summation extends over all possible extensions f_{j}* of f_{j}.

Finally, given a module _{R}S ⊇ M, we define G(S|M), the *Galois group of S over M*, to be the set of all automorphisms of _{R}S
that induce the identify on M.

The first section demonstrates that the injective hull and splitting module have the same closure properties as the algebraic closure and splitting field. For example, it is shown that if M ⊆ _{R}S ⊆ H(M), then S is a splitting module for some family of ideal homomorphisms into M if and only if S is stable under the Galois group of H(M) over M.

As in the case of field theory, the deeper results require a finiteness hypothesis. The main result of § 2 states that if _{R}(R/I) has DCC, then the Galois group of the splitting module of any family of homomorphisms from I to M is solvable.

In the third section we show that if R is noetherian, then Baer's criterion can be improved to state: If _{R}E is an essential extension of _{R}M and every ideal homomorphism from I to M has a root in E, then E is the injective hull of M. We show by an example that the hypothesis "noetherian" cannot be dropped.