Artin-Wedderburn Theorem

The Artin-Wedderburn Theorem is a fundamental result in the field of algebra, specifically in the theory of rings and modules. This theorem provides a complete classification of semisimple rings and has profound implications in representation theory and module theory. The theorem is named after mathematicians Emil Artin and Joseph Wedderburn.

Statement of the Theorem

The Artin-Wedderburn Theorem states that every semisimple ring is isomorphic to a finite direct product of matrix rings over division rings. Formally, if \( R \) is a semisimple ring, then there exist division rings \( D_1, D_2, \ldots, D_n \) and positive integers \( n_1, n_2, \ldots, n_n \) such that:

\[ R \cong \bigoplus_{i=1}^n M_{n_i}(D_i) \]

where \( M_{n_i}(D_i) \) denotes the ring of \( n_i \times n_i \) matrices over the division ring \( D_i \).

Historical Context

The theorem was independently discovered by Emil Artin and Joseph Wedderburn in the early 20th century. Their work laid the groundwork for modern ring theory and module theory. Artin's approach was more general, while Wedderburn's work focused on finite-dimensional algebras over fields.

Semisimple Rings and Modules

A ring \( R \) is called semisimple if it is a direct sum of simple modules. A module \( M \) over a ring \( R \) is simple if it has no proper submodules other than \( 0 \). The concept of semisimplicity is crucial in understanding the structure of rings and modules.

In the context of the Artin-Wedderburn Theorem, a semisimple ring can be decomposed into a direct sum of matrix rings over division rings. This decomposition reveals the internal structure of the ring and allows for a deeper understanding of its properties.

Division Rings

A division ring (or skew field) is a ring in which division is possible, i.e., every non-zero element has a multiplicative inverse. Division rings generalize the concept of fields, allowing for non-commutative multiplication. Examples of division rings include the quaternions and the real numbers.

Matrix Rings

A matrix ring \( M_n(D) \) is the ring of \( n \times n \) matrices over a division ring \( D \). Matrix rings play a crucial role in the Artin-Wedderburn Theorem, as they form the building blocks of semisimple rings. The structure of matrix rings over division rings is well-understood, making them ideal for classifying semisimple rings.

Applications

The Artin-Wedderburn Theorem has numerous applications in various areas of mathematics, including:

**Representation Theory**: The theorem provides a classification of semisimple algebras, which are essential in the study of group representations and Lie algebras.
**Module Theory**: Understanding the structure of semisimple rings helps in the classification of modules over these rings.
**Algebraic Geometry**: The theorem is used in the study of vector bundles and sheaves over algebraic varieties.
**Number Theory**: The classification of semisimple rings has implications in the study of algebraic number fields and local fields.

Proof Outline

The proof of the Artin-Wedderburn Theorem involves several key steps:

1. **Wedderburn's Principal Theorem**: This theorem states that every semisimple algebra over a field is isomorphic to a direct sum of matrix algebras over division rings. 2. **Density Theorem**: This theorem asserts that any simple module over a semisimple ring is isomorphic to a module of the form \( D^n \), where \( D \) is a division ring. 3. **Structure Theorem for Semisimple Rings**: Combining the above results, one can show that any semisimple ring is isomorphic to a direct sum of matrix rings over division rings.

The detailed proof requires a deep understanding of module theory, ring homomorphisms, and isomorphisms.

Examples

To illustrate the Artin-Wedderburn Theorem, consider the following examples:

**Example 1**: The ring of \( 2 \times 2 \) matrices over the real numbers, \( M_2(\mathbb{R}) \), is a semisimple ring. According to the theorem, it is isomorphic to itself, as \( \mathbb{R} \) is a division ring.
**Example 2**: The ring of complex numbers, \( \mathbb{C} \), is a semisimple ring. It is isomorphic to the matrix ring \( M_1(\mathbb{C}) \), as \( \mathbb{C} \) is a division ring.
**Example 3**: The direct sum of two matrix rings, \( M_2(\mathbb{R}) \oplus M_3(\mathbb{C}) \), is a semisimple ring. According to the theorem, it is isomorphic to itself, as both \( \mathbb{R} \) and \( \mathbb{C} \) are division rings.

Generalizations

The Artin-Wedderburn Theorem has been generalized in several ways:

**Infinite-Dimensional Algebras**: The theorem has been extended to certain classes of infinite-dimensional algebras, such as C*-algebras and von Neumann algebras.
**Non-Associative Algebras**: Generalizations to non-associative algebras, such as Jordan algebras and Lie algebras, have been explored.
**Categorical Approaches**: The theorem has been studied from a categorical perspective, leading to insights in category theory and homological algebra.

Related Theorems

Several theorems are closely related to the Artin-Wedderburn Theorem:

**Maschke's Theorem**: This theorem states that the group algebra of a finite group over a field of characteristic not dividing the group order is semisimple.
**Jacobson Density Theorem**: This theorem provides a characterization of simple modules over semisimple rings.
**Wedderburn's Little Theorem**: This theorem classifies finite division rings, showing that every finite division ring is a finite field.