Posted by Jason Polak on 29. December 2016 · Write a comment · Categories: commutative-algebra · Tags:

Let $R$ be a commutative ring. We say that an $R$-algebra $A$ is separable if it is projective as an $A\otimes_R A^{\rm op}$-module. Examples include full matrix rings over $R$, finite separable field extensions, and $\Z[\tfrac 12,i]$ as a $\Z[\tfrac 12]$-algebra.

The 1970 classic Separable Algebras by deMeyer and Ingraham acquaints the reader with this important class of algebras from two viewpoints: the noncommutative one through structure theory and the Brauer group, and the commutative one through Galois theory.

This book accomplished the rare feat of keeping me interested; throughout its pages I found I could apply its results to familiar situations: Why are the only automorphisms of full matrix rings over fields inner? Why are such rings simple? What makes Galois theory tick? Separable Algebras explains with clarity how familiar algebra works through the lens of separable algebras.
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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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Let us recall some classic words:

Our subject starts with homology, homomorphisms, and tensors.

Saunders Mac Lane, in "Homology"

And while Mac Lane's "Homology" and its friend by Cartan and Eilenberg are certainly fairly comprehensive sources of homological algebra, viewpoint shifts in the subject have made more recent approaches desirable. Weibel's 'An Introduction to Homological Algebra' (author website, Amazon), or IHA, is just that: a modern textbook on homological algebra. Aside from a few busy semesters, during the last two years I have been slowly reading it, as I was determined to read this book cover-to-cover. Now that I have finished this, it is my pleasure to write a short review of this book.
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