Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors.
An incomplete list of such contributors includes E. Artin, Richard Brauer, P. M. Cohn, W. R. Hamilton, I. N. Herstein, N. Jacobson, K. Morita, E. Noether, Ø.
It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side.
A module over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
[7][8] This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.
[9] This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings.
More formally, the theorem can be stated as follows: Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U.
If S is the set of the non zero elements of an integral domain, then the localization is the field of fractions and thus usually denoted Frac(R).
There is now a large mathematical theory about it, named microlocalization, connecting with numerous other branches.
Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties.
It is named after Japanese mathematician Kiiti Morita who defined equivalence and a similar notion of duality in 1958.
Further it can be shown that any functor from R-Mod to S-Mod that yields an equivalence is automatically additive.
The Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras.
It arose out of attempts to classify division algebras over a field and is named after the algebraist Richard Brauer.
More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras.
The Ore condition is a condition introduced by Øystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more generally localization of a ring.
A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings.