What follows is adapted from T.Y. Lam’s book *Lectures on modules and rings. *

In this previous post we talked about dependence relations and how they encompass many particular ideas of what it means for some algebraic object to be spanned by a subset of elements. We also noted that the Steinitz replacement axiom fails in general rings. This post shows how to generalise our definitions so that an analogous replacement theorem holds. But first, some definitions!

Given a ring and an -module , call a submodule *essential *if it has nontrivial intersection with every nontrivial submodule of . Of course, an essential submodule must be nonzero. Call uniform if every pair of nonzero submodules of have nontrivial intersection. Equivalently, every submodule of is essential.

Exercise.If is a vector space, a submodule is uniform if and only if it is one dimensional.

It is an idea of A. Goldie (after whom *Goldie rings *are so called) that in order to measure the size of a module, one can try to see how many direct sums of uniform submodules one can fit inside . After all, the dimension of a vector space is precisely this number. One calls this the uniform dimension of the given module. Is this well defined in the general case?

The answer is affirmative, and it is the generalized version of the Steinitz replacement theorem:

Theorem.Assume and are essential submodules of a module and the are uniform. Then .

*Proof *We can safely assume that . The claim is intersects some trivially. Else would be essential in , and the direct sum of such intersections (which is ), would be essential in . It would follow then, by transitivity, that $latex \hat U\cap V$ is essential in , so is, which is impossible. By relabelling we may assume then that . Let , so that is nontrivial (why?). Since is uniform, is essential in and thus in . Since this is contained in , itself is essential in . We have thus replaced by . After steps, we obtain that is essential in , which forces , as desired.

Combining this with the following lemma one obtains our definition of uniform dimension: we say has* uniform dimension* if there is an essential submodule that is a direct sum of uniform submodules of .

Lemma.Suppose that is a direct sum of uniform submodules of a given module and that is maximum among the possible number of uniform direct sumands of uniform submodules one can fit inside . Then is an essential submodule of .

*Proof *This is not completely straightforward, but it is once one proves that the uniform dimension of is the supremum of the number of *direct sumands* one can fit inside . If this weren’t essential, there would exist some nonzero such that $N\cap N’=\varnothing$, and then would contain , which violates maximality.

In order to prove that the uniform dimension of is the supremum of the number of *direct sumands* one can fit inside , we prove:

Proposition.If has finite uniform dimension , then any direct sum of nonzero submodule in has .

*Proof. *Let be an essential submodule of that is a direct sum of uniform submodules, and assume . Then the are nonzero and form a direct sum inside , so we might as well assume that . The same argument as in the proof of the replacement theorem gives a relabeling of the such that . We may then project and obtain an embedding of in $\hat V$. By induction, , so .

Proposition.A module has infinite uniform dimension if and only if it contains an infinite direct summand.

*Proof. *One direction is evident from the previous proposition. Suppose now that does not contain any infinite direct sum. Then every nonzero submodule of contains a uniform submodule. Indeed, if this fails for some nonzero, then in particular is not uniform, so a nonzero direct sum decomposition. Then is not uniform, and it may be decomposed. We then inductively construct an infinite direct sum, which cannot be. We may thus pick a uniform submodule . If is not essential then we may obtain a such that , and by picking some uniform submodule of we may assume is uniform. Continuing, this process must terminate, arriving at some that is essential and a direct sum of uniform submodules.

Corollary.The uniform dimension of a module is the supremum of the number ofdirect sumandsit contains.

Facts.Some happy, some sad:

- Any artinian of noetherian module has finite uniform dimension.
- If a module has finite composition length , it has uniform dimension at most , with equality if and only if this module is semisimple.
- Sadly, finitely generated modules do not necessarily have finite dimension. The direct product is cyclic over itself, but contains a direct sum of ideals, one for each copy of the integers.
- Dimension is additive (meaning the dimension of a direct sum is the direct sum of the dimensions.)
- Dimension is monotone, that is, the dimension of any submodule of is at most that of . Equality is attained if this submodule is essential.
- Sadly, dimension is not additive for exact sequences: as an example, has uniform dimension , so the obvious exact sequence does the job.
- If is a torsion-free module over a commutative domain with field of fractions ; then the dimension of is precisely . More generally, the uniform dimension of the torsion-free quotient of is this dimension.