Simple ring

A nonzero ring is a simple algebra if it contains no two-sided ideals other than the zero ideal and the ring itself.

An immediate example of simple algebras are division algebras, where every nonzero element has a multiplicative inverse, for instance, the real algebra of quaternions. Also, for any ${\displaystyle n\geq 1}$, the algebra of ${\displaystyle n\times n}$ matrices with entries in a division ring is simple. In fact, this characterizes all finite-dimensional simple algebras up to isomorphism, i.e., any simple algebra that is finite-dimensional over its center is isomorphic to a matrix algebra over some division ring. This was proved in 1907 by Joseph Wedderburn in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. Wedderburn's thesis classified simple and semisimple algebras. Simple algebras are building blocks of semisimple algebras: any finite-dimensional semisimple algebra is a Cartesian product, in the sense of algebras, of simple algebras.

Wedderburn's result was later generalized to semisimple rings in the Artin–Wedderburn theorem.

• A central simple algebra (sometimes called Brauer algebra) is a simple finite-dimensional algebra over a field ${\displaystyle F}$ whose center is ${\displaystyle F}$.

Let ${\displaystyle \mathbb {R} }$ be the field of real numbers, ${\displaystyle \mathbb {C} }$ be the field of complex numbers, and ${\displaystyle \mathbb {H} }$ the quaternions.

• Every finite-dimensional simple algebra over ${\displaystyle \mathbb {R} }$ is isomorphic to a matrix ring over ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {C} }$, or ${\displaystyle \mathbb {H} }$. Every central simple algebra over ${\displaystyle \mathbb {R} }$ is isomorphic to a matrix ring over ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {H} }$. These results follow from the Frobenius theorem.
• Every finite-dimensional simple algebra over ${\displaystyle \mathbb {C} }$ is a central simple algebra, and is isomorphic to a matrix ring over ${\displaystyle \mathbb {C} }$.
• Every finite-dimensional central simple algebra over a finite field is isomorphic to a matrix ring over that field.
• For a commutative ring, the four following properties are equivalent: being a semisimple ring; being Artinian and reduced; being a reduced Noetherian ring of Krull dimension 0; and being isomorphic to a finite direct product of fields.

Wedderburn's theorem characterizes simple rings with a unit and a minimal left ideal. (The left Artinian condition is a generalization of the second assumption.) Namely it says that every such ring is, up to isomorphism, a ring of ${\displaystyle n\times n}$ matrices over a division ring.

For the proof, see either Artinian ring#Simple Artinian ring or the proof below.

Let ${\displaystyle D}$ be a division ring and ${\displaystyle M_{n}(D)}$ be the ring of matrices with entries in ${\displaystyle D}$. It is not hard to show that every left ideal in ${\displaystyle M_{n}(D)}$ takes the following form:

${\displaystyle \{M\in M_{n}(D)\mid {\text{the }}n_{1},\dots ,n_{k}{\text{-th columns of }}M{\text{ have zero entries}}\}}$,

for some fixed subset ${\displaystyle \{n_{1},\dots ,n_{k}\}\subseteq \{1,\dots n\}}$. So a minimal ideal in ${\displaystyle M_{n}(D)}$ is of the form

${\displaystyle \{M\in M_{n}(D)\mid {\text{all but the }}k{\text{-th columns have zero entries}}\}}$,

for a given ${\displaystyle k}$. In other words, if ${\displaystyle I}$ is a minimal left ideal, then ${\displaystyle I=M_{n}(D)e}$, where ${\displaystyle e}$ is the idempotent matrix with 1 in the ${\displaystyle (k,k)}$ entry and zero elsewhere. Also, ${\displaystyle D}$ is isomorphic to ${\displaystyle eM_{n}(D)e}$. The left ideal ${\displaystyle I}$ can be viewed as a right module over ${\displaystyle eM_{n}(D)e}$, and the ring ${\displaystyle M_{n}(D)}$ is clearly isomorphic to the algebra of homomorphisms on this module.

The above example suggests the following lemma:

Lemma. ${\displaystyle A}$ is a ring with identity ${\displaystyle 1}$ and an idempotent element ${\displaystyle e}$, where ${\displaystyle AeA=A}$. Let ${\displaystyle I}$ be the left ideal ${\displaystyle Ae}$, considered as a right module over ${\displaystyle eAe}$. Then ${\displaystyle A}$ is isomorphic to the algebra of homomorphisms on ${\displaystyle I}$, denoted by ${\displaystyle \operatorname {Hom} (I)}$.

Proof: We define the "left regular representation" ${\displaystyle \phi \colon A\to \operatorname {Hom} (I)}$ by ${\displaystyle \phi (a)m=am}$ for ${\displaystyle m\in I}$. Then ${\displaystyle \phi }$ is injective because if ${\displaystyle a\cdot I=aAe=0}$, then ${\displaystyle aA=aAeA=0}$, which implies that ${\displaystyle a=a\cdot 1=0}$.

For surjectivity, let ${\displaystyle T\in \operatorname {Hom} (I)}$. Since ${\displaystyle AeA=A}$, the unit ${\displaystyle 1}$ can be expressed as ${\displaystyle \textstyle 1=\sum a_{i}eb_{i}}$. So

${\displaystyle \textstyle T(m)=T(1\cdot m)=T(\sum a_{i}eb_{i}m)=\sum T(a_{i}eeb_{i}m)=\sum T(a_{i}e)eb_{i}m=(\sum T(a_{i}e)eb_{i})m}$.

Since the expression ${\displaystyle \textstyle (\sum T(a_{i}e)eb_{i})}$ does not depend on ${\displaystyle m}$, ${\displaystyle \phi }$ is surjective. This proves the lemma.

Wedderburn's theorem follows readily from the lemma.

Theorem (Wedderburn). If ${\displaystyle A}$ is a simple ring with unit ${\displaystyle 1}$ and a minimal left ideal ${\displaystyle I}$, then ${\displaystyle A}$ is isomorphic to the ring of ${\displaystyle n\times n}$ matrices over a division ring.

One simply has to verify the assumptions of the lemma hold, i.e. find an idempotent ${\displaystyle e}$ such that ${\displaystyle I=Ae}$, and then show that ${\displaystyle eAe}$ is a division ring. The assumption ${\displaystyle A=AeA}$ follows from ${\displaystyle A}$ being simple.

• Simple (algebra)
• Simple universal algebra

• A. A. Albert, Structure of algebras, Colloquium publications 24, American Mathematical Society, 2003, ISBN 0-8218-1024-3. P.37.
• Bourbaki, Nicolas (2012), Algèbre Ch. 8 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-35315-7
• Henderson, D.W. (1965). "A short proof of Wedderburn's theorem". Amer. Math. Monthly. 72: 385–386. doi:10.2307/2313499.
• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4419-8616-0, ISBN 978-0-387-95325-0, MR 1838439
• Lang, Serge (2002), Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0387953854
• Jacobson, Nathan (1989), Basic algebra II (2nd ed.), W. H. Freeman, ISBN 978-0-7167-1933-5