# Dimension zero rings for three types of dimension

Posted by Jason Polak on 20. September 2017 · Write a comment · Categories: homological-algebra, model-theory

There are all sorts of notions of dimension that can be applied to rings. Whatever notion you use though, the ones with dimension zero are usually fairly simple compared with the rings of higher dimension. Here we’ll look at three types of dimension and state what the rings of zero dimension look like with respect to each type. Of course, several examples are included.

All rings are associative with identity but not necessarily commutative. Some basic homological algebra is necessary to understand all the definitions.

## Global Dimension

The left global dimension of a ring $R$ is the supremum over the projective dimensions of all left $R$-modules. The right global dimension is the same with “left” replaced by “right”. And yes, there are rings where the left and right global dimensions differ.

However, $R$ has left global dimension zero if and only if it has right global dimension zero. So, it makes sense to say that such rings have global dimension zero. Here is their characterisation:

A ring $R$ has global dimension zero if and only if it is semisimple; that is, if and only if it is a finite direct product of full matrix rings over division rings.

Examples of such rings are easy to generate by this characterisation:

1. Fields and finite products of fields
2. $M_2(k)$, the ring of $2\times 2$ matrices over a division ring $k$
3. etc.

# Every Set Has a Group Structure Iff Axiom of Choice

Posted by Jason Polak on 15. October 2011 · 1 comment · Categories: model-theory, set-theory · Tags: , ,

Here I explain the proof that in ZF, the axiom of choice (AC) is equivalent to every nonempty set having group structure (GS). I first learned of the nontrivial direction of this argument in this MathOverflow post and as far as I know first appeared in “Some new algebraic equivalents of the axiom of choice” by A. Hajnal and A. Kertész in Publ. Math. Debrecen 19 (1972), pp. 339-340.