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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 since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra .
Successor to Derive. Based on Derive's engine used in TI-89/Voyage 200 and TI-Nspire handheld. Wolfram Alpha. Wolfram Research. 2009. 2013. Pro version: $4.99 / month, Pro version for students: $2.99 / month, ioRegular version: free. Proprietary. Online computer algebra system with step-by step solutions.
In mathematics, the tensor algebra of a vector space V, denoted T(V) or T • (V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product.It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing V, in the sense of the corresponding universal property ...
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Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function u : E1 → E2 between vector spaces is entirely determined by its values on a basis of the vector space E1. The following definition translates this to any category. A concrete category is a category that is equipped with a ...
In algebra, the free product ( coproduct) of a family of associative algebras over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the 's. The free product of two algebras A, B is denoted by A ∗ B. The notion is a ring-theoretic analog of a free product of groups .
An associative R-algebra A (or more simply, an R-algebra A) is a ring A that is also an R -module in such a way that the two additions (the ring addition and the module addition) are the same operation, and scalar multiplication satisfies. ⋅ () = ( ⋅) = ( ⋅) for all r in R and x, y in the algebra.
Algebra is the branch of mathematics that studies algebraic operations [a] and algebraic structures. [2] An algebraic structure is a non-empty set of mathematical objects, such as the real numbers, together with algebraic operations defined on that set, such as addition and multiplication. [3] Algebra explores the laws, general characteristics ...
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