Simple algebra

In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field.

The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this field. So, simple algebra and ''simple ring'' are synonyms.

Several references (e.g., Lang (2002) or Bourbaki (2012)) require in addition that a simple ring be left or right Artinian (or equivalently semi-simple). Under such terminology a non-zero ring with no non-trivial two-sided ideals is called quasi-simple.

Rings which are simple as rings but are not a simple module over themselves do exist: a full matrix ring over a field does not have any nontrivial ideals (since any ideal of $M_n(R)$ is of the form $M_n(I)$ with $I$ an ideal of $R$), but has nontrivial left ideals (for example, the sets of matrices which have some fixed zero columns).

According to the Artin–Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.

An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra.

Characterization

A ring is a simple algebra if it contains no non-trivial two-sided ideals.

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, one can show that the algebra of $n \times n$ 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 semi-simple algebras: any finite-dimensional semi-simple 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.

Examples

* A central simple algebra (sometimes called Brauer algebra) is a simple finite-dimensional algebra over a field $F$ whose center is $F$.
Let $\mathbb$ be the field of real numbers, $\mathbb$ be the field of complex numbers, and $\mathbb$ the quaternions.
* Every finite-dimensional simple algebra over $\mathbb$ is isomorphic to a matrix ring over $\mathbb$, $\mathbb$, or $\mathbb$. Every central simple algebra over $\mathbb$ is isomorphic to a matrix ring over $\mathbb$ or $\mathbb$. These results follow from the Frobenius theorem.
* Every finite-dimensional simple algebra over $\mathbb$ is a central simple algebra, and is isomorphic to a matrix ring over $\mathbb$.
* 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

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 $n \times n$ matrices over a division ring.

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

:$\$,

for some fixed subset $\ \subseteq \$. So a minimal ideal in $M_n(D)$ is of the form

:$\$,

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

The above example suggests the following lemma:

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

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

For surjectivity, let $T \in \operatorname(I)$. Since $AeA = A$, the unit $1$ can be expressed as $\textstyle 1 = \sum a_i e b_i$. So

:$\textstyle T(m) = T(1 \cdot m) = T(\sum a_i e b_i m) = \sum T(a_i e e b_i m) = \sum T(a_i e) e b_i m = (\sum T(a_i e) e b_i) m$.

Since the expression $\textstyle (\sum T(a_i e) e b_i)$ does not depend on $m$, $\phi$ is surjective. This proves the lemma.

Wedderburn's theorem follows readily from the lemma.

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

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

See also

* Simple (algebra)
* Simple universal algebra

References

* A. A. Albert, ''Structure of algebras'', Colloquium publications 24, American Mathematical Society, 2003, . P.37.
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* {{Citation , last1=Jacobson , first1=Nathan , author1-link=Nathan Jacobson , title=Basic algebra II , publisher=W. H. Freeman , edition=2nd , isbn=978-0-7167-1933-5 , year=1989
Ring theory