In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. The number 0 is an important concept in Mathematics. Zero module In Mathematics, the zero module is the module consisting of only A simple ring can always be considered as a simple algebra. In Mathematics, specifically in Ring theory, an algebra is simple if it contains no non-trivial two-sided ideals and the set {
According to the Artin-Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In Abstract algebra, the Artin–Wedderburn theorem is a Classification theorem for semisimple rings. In Abstract algebra, an Artinian ring is a ring that satisfies the Descending chain condition on ideals. In Abstract algebra the matrix ring M( n, R) is the set of all n × n matrices over an arbitrary ring In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible 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. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician
Any quotient of a ring by a maximal ideal is a simple ring. In Mathematics, more specifically in Ring theory, a maximal ideal is an ideal which is maximal (with respect to set inclusion amongst all proper ideals In particular, a field is a simple ring. A ring R is simple if and only its opposite ring Ro is simple. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real
An example of a simple ring that is not a matrix ring over a division ring is the Weyl algebra. In Abstract algebra, the Weyl algebra is the ring of Differential operators with Polynomial coefficients (in one variable
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. . . nk-th columns of M have zero entries},
for some fixed {n1,. . . ,nk} ⊂ {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 e(M(n,D))e. The left ideal I can be viewed as a right-module over e(M(n,D))e, and the ring M(n,D) is clearly isomorphic to the algebra of homorphisms 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 a = a · 1 = 0. For surjectivity, let T ∈ Hom(I). Since AeA = A, the unit 1 can be expresses as 1 = ∑aiebi. So
Since the expression [∑T(aie)ebi] does not depend on m, Φ is surjective. This proves the lemma.
- T(m) = T(1·m) = T(∑aiebim) = ∑ T(aieebim) = ∑ T(aie) ebim = [ ∑T(aie)ebi]m.
Wedderburn's theorem follows readily from the lemma.
Theorem (Wedderburn). If A is a semisimple 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 A = AeA. A being simple also implies eAe is a division ring.
See also
References
- D. In Mathematics, the term simple is used to describe an Algebraic structures which in some sense cannot be divided by a smaller structure of the same type W. Henderson, A short proof of Wedderburn's theorem, Amer. Math. Monthly 72 (1965), 385-386.
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