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 or Bourbaki ) require in addition that a simple ring be left or right Artinian. 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 is of the form Mn, but has nontrivial left ideals. 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 ringRo 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 is a simple commutative ringif and only if is a field. This is because if is a commutative ring, then you can pick a nonzero element and consider the ideal. Then since is simple, this ideal is the entire ring, and so it contains 1, and therefore there is some element such that, and so is a field. Conversely, if is known to be a field, then any nonzero ideal will have a nonzero element. But since is a field, then and so, and so.
Simple algebra
An algebra is simple if it contains no non-trivial two-sided ideals and the multiplication operation is not zero. The second condition in the definition precludes the following situation; consider the algebra with the usual matrix operations: 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.
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.
Wedderburn's theorem characterizes simple rings with a unit and a minimal left ideal. Namely it says that every such ring is, up to isomorphism, a ring of -matrices over a division ring. Let D be a division ring and be the ring of matrices with entries in D. It is not hard to show that every left ideal in takes the following form: for some fixed ⊆. So a minimal ideal in is of the form for a given k. In other words, if I is a minimal left ideal, then, where e is the idempotent matrix with 1 in the entry and zero elsewhere. Also, D is isomorphic to. The left ideal I can be viewed as a right module over, and the ring 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. 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.
Proof: We define the "left regular representation" for. Φ is injective because if, then, which implies that. For surjectivity, let. Since, the unit 1 can be expressed as. So Since the expression does not depend on m, Φ is surjective. This proves the lemma.
Wedderburn's theorem follows readily from the lemma.
Theorem. If A is a simple ring with unit 1 and a minimal left ideal I, then A is isomorphic to the ring of -matrices over a division ring.
One simply has to verify the assumptions of the lemma hold, i.e. find an idempotent e such that, and then show that eAe is a division ring. The assumption follows fromA being simple.
