Category Archives: commutative-algebra

## Ideal in a union of ideals

Suppose $I$ is an ideal in a ring $R$ and $J,K$ are ideals such that $I\subseteq J\cup K$. Then either $I\subseteq J$ or $I\subseteq K$. Indeed, suppose that there is some $x\in I$ such that $x\not\in J$. If $y\in I$ is arbitrary and $y\not\in K$ then $x + y$ is in neither $J$ nor $K$. […]

## Degrees of some permutation polynomials

Let $\F_q$ be a finite field. For any function $f:\F_q\to \F_q$, there exists a polynomial $p\in \F_q[x]$ such that $f(a) = p(a)$ for all $a\in \F_q$. In other words, every function from a finite field to itself can be represented by a polynomial. In particular, every permutation of $\F_q$ can be represented by a polynomial. […]

## Zero divisor graph of a commutative ring

Let $R$ be a commutative ring. The zero divisors of $R$, which we denote $Z(R)$ is the set-theoretic union of prime ideals. This is just because in any commutative ring, the set of subsets of $R$ that can be written as unions of prime ideals is in bijection with the saturated multiplicatively closed sets (the […]

## Working with group rings in Sage

Let $\Z[\Z/n]$ denote the integral group ring of the cyclic group $\Z/n$. How would you create $\Z[\Z/n]$ in Sage so that you could easily multiply elements? First, if you’ve already assigned a group to the variable ‘A’, then

will give you the corresponding group ring and store it in the variable ‘R’. The first […]

## What’s in all powers of a principal prime?

Let $R$ be a commutative ring and $(p)$ be a principal prime ideal. What can be said about the intersection $\cap_{k=1}^\infty (p)^k$? Let’s abbreviate this $\cap (p)^k$ (I like to use the convention that when limits are not specified, then the operation like intersection is taken over all possible indices). Let’s try an example. For […]

## Poset of prime ideals

For a commutative ring, what does the partially ordered set (=poset) of primes look like? I already talked a little about totally ordered sets of primes, but what about in general? For a general partially ordered set $S$ there are two immediate questions that come to mind: Does there exist a commutative ring whose poset […]

## 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 $k$ […]

## When the set of prime ideals is linearly ordered

Imposing structure on the poset of prime ideals of a ring $R$ is one way to gain a hold onto its structure. The poset of prime ideals of $R$ is simply a fancy term for the set of prime ideals of $R$, partially ordered by inclusion. Usually this set is not totally ordered: in the […]

## 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 […]

## Number of irreducible polynomials over a finite field

Over a finite field, there are of course only finitely many irreducible monic polynomials. But how do you count them? Let $q = p^n$ be a power of a prime and let $N_q(d)$ denote the number of monic irreducible polynomials of degree $d$ over $\F_q$. The key to finding $N_q(d)$ is the following fact: the […]