We have received a new set of donations, again from a live-TeXed course. One thing that I hadn’t seen before is a theorem of McCoy that gives a criterion (in the “density” of certain ideals generated by minors) of when a matrix mapping of free modules is injective.

In addition, various edits keep being made, slowly. There have been several articles in the Monthly that give short semi-expository pieces of mathematics; a brief account of some of them have appeared in the book. Here is one, giving a short proof of the following lemma: a field extension which is finitely generated as an algebra is algebraic.

To see this, one can use the following argument of McCabe. Let be an extension finitely generated as an algebra. There is a subring over which $k’$ is algebraic. The finite generation hypothesis implies that we can find a localization over which $k’$ is even integral. But if one has an integral extension and the top ring is a field, then so is the bottom one; this implies is a field. So must lie in every maximal ideal, meaning it is zero, contradiction.