Simple ring

In abstract algebra, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself.

One should notice that 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.

A simple ring can always be considered as a simple algebra over its center. Rings which are simple as rings but not as modules do exist: the 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 (namely, 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.

Any quotient of a ring by a maximal two-sided ideal is a simple ring. In particular, a field is a simple ring. In fact a division ring is also a simple ring. A ring is simple if and only its opposite ring R^o is simple.

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

Furthermore, a ring $R$ is a simple commutative ring if and only if $R$ is a field. This is because if $R$ is a commutative ring, then you can pick a nonzero element $x\in R$ and consider the ideal $\{xr:r\in R\}$ . Then since $R$ is simple, this ideal is the entire ring, and so it contains 1, and therefore there is some element $y\in R$ such that $xy=1$ , and so $R$ is a field. Conversely, if $R$ is known to be a field, then any nonzero ideal $I\subseteq R$ will have a nonzero element $i\in I$ . But since $R$ is a field, then $i^{-1}\in R$ and so $i^{-1}i=1\in I$ , and so $I=R$ .

Simple algebra

An algebra is simple if it contains no non-trivial two-sided ideals and the multiplication operation is not zero (that is, there is some a and some b such that ab ≠ 0).

The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations:

\left\{\left.{\begin{bmatrix}0&\alpha \\0&0\\\end{bmatrix}}\,\right|\,\alpha \in \mathbb {C} \right\}

This is a one-dimensional algebra in which the product of any two elements is zero. This condition ensures that the algebra has a minimal nonzero left ideal, which simplifies certain arguments.

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 × 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 finite-dimensional simple algebra is isomorphic to a matrix algebra over some division ring. This result was given in 1907 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 R be the field of real numbers, C be the field of complex numbers, and H the quaternions.

Every finite-dimensional simple algebra over R is isomorphic to a matrix ring over R, C, or H. Every central simple algebra over R is isomorphic to a matrix ring over R or H. These results follow from the Frobenius theorem.
Every finite-dimensional simple algebra over C is a central simple algebra, and is isomorphic to a matrix ring over 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; 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 × 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:

{M ∈ M_n(D) | The n₁...n_k-th columns of M have zero entries},

for some fixed {n₁, ..., n_k} ⊆ {1, ..., n}. So a minimal ideal in M_n(D) is of the form

{M ∈ M_n(D) | All but the k-th columns have zero entries},

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 Hom(I).

Proof: We define the "left regular representation" Φ : A → Hom(I) by Φ(a)m = am for m ∈ I. Φ is injective because if a ⋅ I = aAe = 0, then aA = aAeA = 0, which implies that a = a ⋅ 1 = 0.

For surjectivity, let T ∈ Hom(I). Since AeA = A, the unit 1 can be expressed as 1 = ∑a_ieb_i. So

T(m) = T(1⋅m) = T(∑a_ieb_im) = ∑ T(a_ieeb_im) = ∑ T(a_ie) eb_im = [∑T(a_ie)eb_i]m.

Since the expression [∑T(a_ie)eb_i] does not depend on m, Φ 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 × 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.

References

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

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Simple ring

Simple algebra

Examples

Wedderburn's theorem

See also

References