Category Archives: commutative-algebra

Replacing two idempotents with one

Let $R$ be a commutative ring. Two idempotents $e$ and $f$ are called orthogonal if $ef = 0$. The archetypal example is $(0,1)$ and $(1,0)$ in a product ring $R\times S$. Let $e$ and $f$ be orthogonal idempotents. Then the ideal $(e,f)$ is equal to the ideal $(e + f)$. To see, this first note […]

Extensions of Finite Rings are Integral

Here's a classic definition: let $R\subseteq S$ be commutative rings. An element $s\in S$ is called integral over $R$ if $f(s)=0$ for some monic polynomial $f\in R[x]$. It's classic because appending the solutions of polynomials to base rings goes way back to the ancient pasttime of finding solutions to polynomial equations. For example, consider $\Z\subseteq […]

Non-unique Factorisation: Part 2

We are continuing the series on non-unique factorisation. For a handy table of contents, visit the Post Series directory. In Part 1 of this series, we introduced for a commutative ring three types of relations: Associaties: $a\sim b$ means that $(a) = (b)$ Strong associates: $a\approx b$ means that $a = ub$ for $u\in U(R)$ […]

Non-unique Factorisation: Part 1

If $F$ is a field then the polynomial ring $F[x]$ is a unique factorisation domain: that is, every nonunit can be written uniquely as a product of irreducible elements up to a unit multiple. So in $\Q[x]$ for example, you can be sure that the polynomial $x^2 – 2 = (x-2)(x+2)$ can't be factored any […]

More about Ext Calculations with Regular Sequences

This post is a continuation of this previous one, though I repeat the main definitions for convenience. Let $R$ be a commutative ring and $A$ and $R$-module. We say that $x_1,\dots,x_n\in R$ is a regular sequence on $A$ if $(x_1,\dots,x_n)A\not = A$ and $x_i$ is not a zero divisor on $A/(x_1,\dots,x_{i-1})A$ for all $i$. Last […]

Paper Announcement: Separable Polynomials in Z/n[x]

I'd like to invite readers of this blog to download my latest paper, to appear in the Canadian Mathematical Bulletin: Counting Separable Polynomials in Z/n[x] What is this paper about? It uses the theory of separable algebras to study separable polynomials in $\Z/n[x]$, which extends the usual definition of separability for polynomials over a field. […]

Regular Sequences and Ext Calculations

Let $R$ be a commutative ring and $A$ and $R$-module. We say that $x_1,\dots,x_n\in R$ is a regular sequence on $A$ if $(x_1,\dots,x_n)A\not = A$ and $x_i$ is not a zero divisor on $A/(x_1,\dots,x_{i-1})A$ for all $i$. Regular sequences are a central theme in commutative algebra. Here's a particularly interesting theorem about them that allows […]