Ring Theory and Commutative Algebra

Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field.

to the product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are

of the interpretation of algebraic geometry in terms of commutative algebra. In particular, the basis theorem implies that

is a theorem that establishes a fundamental relationship between geometry and algebra. It was proven by David Hilbert in his second

In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull

In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma - also known as the Krull-Azumaya

to as a theorem rather than a lemma) is a result of commutative algebra, introduced by Emmy Noether in 1926. It states

In ring theory, a branch of mathematics, the Skolem-Noether

is the starting point of the use of homological methods in commutative algebra and algebraic geometry.

In commutative algebra, Krull's principal ideal theorem, named after

Artin-Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis

In commutative algebra, the Krull-Akizuki theorem states the following:

In commutative algebra, a branch of mathematics, going up and going

In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson

mathematics, the Lasker-Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal

by Amitsur and Levitsky (1950). In particular matrix rings are polynomial identity rings such that the smallest