3

I was tasked with finding an example of a ring that is Artinian only on one side. Checking the Database of Ring Theory yielded this result:

Let $σ:k→k$ be a field endomorphism of a countable field $k$ such that $∞=[k:σ(k)]>1$. $k[x;σ]$ is the twisted polynomial ring where $xa:=σ(a)x$ for all $a \in k$. The ring is $k[x;σ]/(x^2)$.

However, I am not familiar with the notation $[ k : σ(k) ]$. It seems similar to cosets from group theory, so I was assuming it would refer to the size of the quotient of $k$ by the image of $σ$, but I can't see how that would work in this context, without making $σ=0$.

Martim Pinto Paiva
  • 59
  • 3

1 Answers1

2

$[k:\sigma(k)]$ is the degree of a field extension.

Note that $\sigma(k)$ is a subfield of $k$. Then $k$ automatically becomes a vector space over $\sigma(k)$. This works similarly to $\mathbb{C}$ being an $\mathbb{R}$-vector space. Now $[k:\sigma(k)]$ is the dimension of $k$ as a $\sigma(k)$-vector space.

Tzimmo
  • 1,299