ASSOCIATIVE ALGEBRA Meaning and
Definition
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An associative algebra is a mathematical structure that combines properties of a vector space and a ring. It is a triple (A, +, •), where A is a set, + is an addition operation, and • is a multiplication operation.
The set A is equipped with an addition operation (+) that satisfies the commutative, associative, and distributive laws. This makes A a vector space over a certain field. The addition operation allows for the combining of elements in A.
The multiplication operation (•) follows the associative law, meaning that the product of three elements is independent of the way they are grouped. This operation also satisfies the distributive law with respect to addition.
Moreover, an associative algebra is equipped with a scalar multiplication operation, allowing for multiplication between elements of the algebra and a scalar from the underlying field. This scalar multiplication operation should adhere to the rules of linearity.
Associative algebras play a crucial role in various branches of mathematics and theoretical physics. They have applications in abstract algebra, representation theory, and quantum mechanics, among others. Matrix algebras, polynomial algebras, and group algebras are examples of associative algebras commonly encountered in mathematics.
Understanding the properties and structure of associative algebras leads to a better comprehension of many mathematical concepts and enables the development of efficient algorithms and mathematical models in diverse areas of research.
Etymology of ASSOCIATIVE ALGEBRA
The term "associative algebra" is derived from the combination of two words: "associative" and "algebra".
The word "associative" is derived from the Latin verb "associare", which means "to join together". It became part of the English language through the Middle French word "associer" and the Old French word "associer". In mathematics, "associative" refers to a property or operation that is independent of the grouping of elements or terms. In other words, the order of elements being combined or the parentheses used in an equation will not affect the result.
The word "algebra" has roots in Arabic mathematics. The term "al-jabr" was used by the mathematician Muhammad ibn Musa al-Khwarizmi in the 9th century, and it meant "reunion of broken parts" or "restoration".
