Chains of Modules, An Introduction

Introduction

Let M be any module, or a left module if you prefer. A chain H of submodules is ascending if H_i ⊂H_i+1, and descending if H_i ⊃ H_i+1. In other words, the submodules are getting larger (ascending) or smaller (descending).

If you like set theory, think of a chain as a map from the ordinals into the submodules of M, where the submodules increase or decrease as the ordinals advance. This allows you to build uncountable chains of submodules. Well ok - most of us will never do that, so let's move on.

Noether (biography) investigated modules with no infinite ascending chains; and these modules are now called noetherian. Do read her biography; it wasn't easy for a Jewish woman to succeed in mathematics, in Germany, in the beginning of the 20^th century.

At approximately the same time, Artin (biography) explored modules with no infinite descending chains, and these modules are now called artinian.

Another synonym for noetherian is "ascending chain condition", or acc. as you might guess, an artinian module has the "descending chain condition", or dcc. We will usually use the words noetherian and artinian.

If R is a ring, R is a left R module, and the submodules are the ideals. Thus a noetherian ring has no infinite ascending chains of ideals, and an artinian ring has no infinite descending chains of ideals.

A division ring is a rather trivial example. It only has two ideals, 0 and the entire ring. This is obviously noetherian and artinian.