Posted by Jason Polak on 31. January 2014 · Write a comment · Categories: commutative-algebra · Tags: ,

P1000536It was more than a year ago that I opened a package that I got in the mail, taking out this green ex-library hardcover in excellent condition. Now, I honestly can’t remember what prompted me to order it (perhaps it was the author’s name), but I remember reading the first few sections and feeling that it would be worthwhile to spend some time over its pages to learn more about those primes. Unfortunately, I didn’t manage to keep reading at the time, but a few months ago I decided to push through this volume with a little spare time I had, and this became my first real serious conversation with the zero divisor.

Certainly, no algebraist can ever escape the grasp of the zero divisor. In a ring $ R$, a nonzero element $ r\in R$ is called a zero divisor if $ rs = 0$ for some nonzero $ s\in R$. In even basic questions on ring theory, zero divisors are bound to be lurking. Our topic today is commutative rings, so we’ll assume from now on that $ R$ is commutative. The rings probably easiest to understand, at least if we’re not considering relations to other rings, are fields. If $ R$ is not a field, then it has an ideal $ I$ that is not prime so $ R/I$ already has zero divisors. So even if $ R$ is a domain, some of its quotients will not be as long as $ R$ is not a field.
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Posted by Jason Polak on 06. May 2012 · 2 comments · Categories: commutative-algebra, modules · Tags: , , , ,

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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