Simple algebra
Encyclopedia
In mathematics
, specifically in ring theory
, an algebra
is simple if it contains no non-trivial two-sided ideal
s and the set {ab | a, b are elements of the algebra} ≠ {0}.
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 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 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 algebra
s. 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
.
, an abstract algebra A is called "simple" if and only if
it has no nontrivial congruence relation
s, or equivalently, if every homomorphism with domain A is either injective or constant.
As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
, an algebra
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
is simple if it contains no non-trivial two-sided ideal
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s and the set {ab | a, b are elements of the algebra} ≠ {0}.
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 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
Matrix algebra
Matrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...
over some division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
. This result was given in 1907 by Joseph Wedderburn
Joseph Wedderburn
Joseph Henry Maclagan Wedderburn was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras...
in his doctoral thesis, On hypercomplex numbers, which appeared in the Proceedings of the London Mathematical Society. Wedderburn's thesis classified simple and semisimple algebra
Semisimple algebra
In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
s. 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
Artin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
.
Examples
- A central simple algebraCentral simple algebraIn ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...
(sometimes called Brauer algebra) is a simple finite dimensional algebra over a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F whose center is F.
Simple universal algebras
In universal algebraUniversal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, an abstract algebra A is called "simple" if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
it has no nontrivial congruence relation
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...
s, or equivalently, if every homomorphism with domain A is either injective or constant.
As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the notion from ring theory: a ring is simple in the sense that it has no nontrivial ideals if and only if it is simple in the sense of universal algebra.