Editing Ring theory (section)

===Structure of noncommutative rings===
{{main|Noncommutative ring}}

The structure of a [[noncommutative ring]] is more complicated than that of a commutative ring. For example, there exist [[Simple ring|simple]] rings that contain no non-trivial proper (two-sided) ideals, yet contain non-trivial proper left or right ideals. Various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the [[nilradical of a ring]], the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all {{nowrap|''n'' × ''n''}} matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.

The concept of the [[Jacobson radical]] of a ring; that is, the intersection of all right (left) [[Annihilator (ring theory)|annihilators]] of [[Simple module|simple]] right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also a fact that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; irrespective of whether the ring is commutative.

Noncommutative rings are an active area of research due to their ubiquity in mathematics. For instance, the ring of ''n''-by-''n'' [[Matrix (mathematics)|matrices over a field]] is noncommutative despite its natural occurrence in [[geometry]], [[physics]] and many parts of mathematics. More generally, [[endomorphism ring]]s of abelian groups are rarely commutative, the simplest example being the endomorphism ring of the [[Klein four-group]].

One of the best-known strictly noncommutative ring is the [[quaternions]].