Conventions and definitions: Rings are unital and not necessarily commutative. Modules over rings are left modules. A local ring is a ring in which the set of nonunits form an ideal. A module is called projective if it is a direct summand of a free module.
Today I shall share with you the wonderful result that any projective module over a local ring is free. We shall follow Kaplansky (reference given below), who first proved this result.
Now modules are in fact my favourite mathematical objects. They are like vector spaces, except that they are interesting. Of course, this “interesting” can be irksome if one has to solve a problem and these interesting properties throw a wrench in the works. However, by themselves modules are certainly curious creatures worthy of intense and gruelling analysis!
Of course, when the idea of a module was first conceived, mathematicians attempted to port all kinds of ideas from vector spaces into the world of modules. Some, like the direct sum construction, worked flawlessly. Other concepts such as rank, fortunately or unfortunately depending on your perspective, did not turn out so well (think about it: if everything worked well with modules then there’d be much less interesting math).
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of order 
for 
is the symmetry group of the regular Euclidean 
-gon. Any dihedral group is generated by a reflection and a certain rotation. Moreover, in Part 1 we gave two other descriptions of the dihedral group 
as a 
module via 
, then 
, which was the second description.
, the dihedral group 
in a bit more detail. By labelling the vertices of an equilateral triangle with 
, we can deduce that 
has order six. In fact, any fixed letter can be rotated to any given position, and then the remaining two letters can be permuted via a reflection, so all of the 
possible configurations obtainable with reflections and rotations.
and 
. Both are vector spaces over the finite field 
so to show that they are isomorphic, we need to show that their respective bases have the same cardinality. The vector space on the right is written as a direct summand and so we can see that its basis must have size 
. On the other hand, the vector space 
Modules
and 
aren’t vector spaces any more, and the previous paragraph would be rife with errors. But they certainly are abelian groups, and they have a bit more spice than those vector spaces. So are they isomorphic? They do have the same cardinality. Fortunately for us, Baer (“Abelian Groups without Elements of Finite Order”, Duke Math J. 3 (1937), pp. 88-122) answered this question in the negative (fortunately, because otherwise abelian groups would be less exciting). In fact, this question is particularly interesting to me because I had wondered about it a few months ago, and now I have the answer, thanks to Faith’s book for the references.
matrix 
with real entries, is it possible to make the sum of each row and each column nonnegative just by multiplying rows and columns by 
? In other words, you are allowed to multiply any row or column by 
are measurable subsets of the real line with positive measure, then 
contains an interval. We shall continue now to prove this again using a different, less involved method. This solution is based on a hint in Problem 19 of Chapter 9 in Rudin’s book.
.
is such that 
contains an interval, yet 
, and 
does not contain an interval. In fact, if we prove the above then we shall have another proof that 
denotes the Lebesgue measure.