Free product of associative algebras information

In algebra, the free product (coproduct) of a family of associative algebras A i , i ∈ I {\displaystyle A_{i},i\in I} over a commutative ring R is the...

article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification...

mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is...

property. The free algebra functor is left adjoint to the forgetful functor from the category of R-algebras to the category of sets. Free algebras over division...

the category of algebras is a free product of algebras.) Direct products are commutative and associative up to natural isomorphism, meaning that it doesn't...

algebra may or may not be associative, leading to the notions of associative algebras and non-associative algebras. Given an integer n, the ring of real...

algebras, as distinct from symplectic Clifford algebras. A Clifford algebra is a unital associative algebra that contains and is generated by a vector space...

algebras of pseudo-differential operators acting on spaces of distributions), the term operator algebra is usually used in reference to algebras of bounded...

saying that polynomial rings are free commutative algebras, since they are free objects in the category of commutative algebras. Similarly, a polynomial ring...

isomorphic to the product of the quotient rings R / In, n = 1, ..., k. An associative algebra A over a commutative ring R is a ring itself. If I is an ideal in A...

z]]} . Nonetheless, much of the terminology for associative rings and algebras (and also for groups) has analogs for Lie algebras. A Lie subalgebra is a...

of these functions commute, the functions also commute: a times b equals b times a. It is remarkable that viewing noncommutative associative algebras...

category of (noncommutative) R-algebras, the coproduct is a quotient of the tensor algebra (see free product of associative algebras). In the case of topological...

to that of associative algebras. Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras. Graded algebra: a graded...

category R-Alg whose objects are all R-algebras and whose morphisms are R-algebra homomorphisms. The category of rings can be considered a special case...

sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of T(V)...

giving a ring homomorphism R → End(M). A unital algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism...

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations...

isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is...

assumed to be commutative or associative. A semifield that is associative is a division ring, and one that is both associative and commutative is a field...

the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It...

Lie algebras. In Lie algebras, the multiplication satisfies Jacobi identity instead of the associative law; this allows abstracting the algebraic nature...

algebra or Grassmann algebra of a vector space V {\displaystyle V} is an associative algebra that contains V , {\displaystyle V,} which has a product...

include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in...