Free algebra

In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring (which may be regarded as a free commutative algebra).

Definition

For R a commutative ring, the free (associative, unital) algebra on n indeterminates {X₁,...,X_n} is the free R-module with a basis consisting of all words over the alphabet {X₁,...,X_n} (including the empty word, which is the unity of the free algebra). This R-module becomes an R-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words:

and the product of two arbitrary elements is thus uniquely determined (because the multiplication in an R-algebra must be R-bilinear). This R-algebra is denoted R⟨X₁,...,X_n⟩. This construction can easily be generalized to an arbitrary set X of indeterminates.

In short, for an arbitrary set , the free (associative, unital) R-algebra on X is

with the R-bilinear multiplication that is concatenation on words, where X* denotes the free monoid on X (i.e. words on the letters X_i), denotes the external direct sum, and Rw denotes the free R-module on 1 element, the word w.

Algebra

Algebra (from Arabic "al-jabr" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Middle East, by mathematicians such as al-Khwārizmī (780 – 850) and Omar Khayyam (1048–1131).