What is a free module?

Among modules over a commutative ring R, the module R^n has a nice feature: homomorphisms from R^n to another module M are entirely determined by where they send the n elements (1,0,\dots,0), (0,1,0,\dots, 0), up to (0,\dots,0,1). Moreover, any choice of where we want those basis elements to be sent in M extends uniquely to a homomorphism R^n\to M. Thus homomorphisms R^n\to M correspond bijectively to n-tuples of elements of M. This is an example of a universal property: R^n is the universal module equipped with an n-tuple of elements, also called the free module on n elements.

What about the free module on an infinite collection of elements? Continue reading

What is a universal property?

The following three constructions have something in common:

  • Kernels: If f:G\to H and g:H\to K are two group homomorphisms, then the composite g\circ f is the trivial homomorphism G\to K if and only if the image of f is contained in the kernel of g.
  • Polynomial rings: If A is any \mathbb{R}-algebra, then an \mathbb{R}-algebra homomorphism \mathbb{R}[x]\to A is entirely determined by where it sends x.
  • Topological products: The product topology on a product of topological spaces \prod_{n=1}^\infty X_n is a little counterintuitive—at first glance, the box topology seems more natural—but the product topology has the property that a function (f_1,f_2,\dots):A\to\prod_{n=1}^\infty X_n is continuous if and only if each component map f_n:A\to X_n is continuous.

Do you already see the similarity? Let me rephrase them to make it more obvious:

Continue reading

Interlude: principal ideal rings

Recently I mentioned that Noetherian rings can be characterized either by the fact that their finitely generated modules don’t have non-finitely generated submodules, or by the a priori weaker condition that their ideals are all finitely generated. In other words, if you want a counterexample to the statement that a submodule of a finitely generated module M is again finitely generated, it’s sufficient to consider the case M = R.

I also claimed that the function \mu_R sending an R-module to its minimum number of generators is not generally increasing: for R-modules N\subseteq M we don’t necessarily have \mu_R(N)\leq \mu_R(M), even when R is Noetherian. For example, over R = \mathbb{C}[x,y], the ideal I = (x,y)\subseteq R has \mu_R(I)=2, but \mu_R(R) = 1.

These two facts led me to ask the following question:

If we want a commutative ring R to have the property that \mu_R is increasing, is it again sufficient to look for counterexamples among the ideals of R? In other words, are the rings whose ideals can all be generated by single elements (the principal ideal rings) exactly the rings with this property?

Continue reading

What is an exact sequence?

I’ve heard it said that the concept of modules was invented to put ideals of rings on the same footing as quotient rings. Both the inclusion of an ideal I into a ring R and the quotient map R \to R/I are homomorphisms of R-modules. In fact, these two homomorphisms have a special relationship: the elements of R that are sent to zero in R/I are exactly those in the image of I\to R.

Continue reading

What is a Noetherian ring?

Noetherianness is about making sure small modules don’t have big parts. But how do you measure how big a module is? Remember that while every vector space has a dimension—the size of a basis—most modules don’t have a basis. We can still talk about a generating set for an R-module M, though, like \{1, 1/2, 1/3, \dots\} for the \mathbb{Z}-module \mathbb{Q}: a subset of M for which every element can be written as an R-linear combination of those generating elements.

Question: For a vector space, its dimension is its minimum number of generators. Can we use the minimum number of generators as a measure of the size of a module in general?

Continue reading