# Tag Archives: example

## Abelian categories: examples and nonexamples

I've been talking a little about abelian categories these days. That's because I've been going over Weibel's An Introduction to Homological Algebra. It's a book I read before, and I still feel pretty confident about the material. This time, though, I think I'm going to explore a few different paths that I haven't really given […]

## Non-Noetherian domain but finitely generated ideals principal

A finitely-generated module over a principal ideal domain is always isomorphic to $R^n\oplus R/a_1\oplus\cdots\oplus R/a_n$ where $n$ is a nonnegative integer and $a_i\in R$ for $i=1,\dots,n$. This is called the structure theorem for modules over a principal ideal domain. Examples of principal ideal domains include fields, $\Z$, $\Z[\sqrt{2}]$, and the polynomial ring $k[x]$ when $k$ […]

## Semisimple and Jacobson Semisimple

Let $R$ be an associative ring with identity. The Jacobson radical ${\rm Jac}(R)$ of $R$ is the intersection of all the left maximal ideals of $R$. So, ${\rm Jac}(R)$ is a left ideal of $R$. It turns out that the Jacobson radical of $R$ is also the intersection of all the right maximal ideals of […]

## Highlights in Linear Algebraic Groups 10: G/B is Projective

In Highlights 9 of this series, we showed that for an algebraic group $G$ and a closed subgroup $H\subseteq G$, we can always choose a representation $G\to\rm{GL}(V)$ with a line $L\subseteq V$ whose stabiliser is $H$. In turn, this allows us to identify the quotient $G/H$ with the orbit […]

## Highlights in Linear Algebraic Groups 7: Representations II

In the previous post, we saw that if $G\times X\to X$ is an algebraic group acting on a variety $X$ and $F\subseteq k[X]$ is a finite-dimensional subspace then there exists a finite dimensional subspace $E\subseteq k[X]$ with $E\supseteq F$ such that $E$ is invariant under translations. Recall that if […]