algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

, the free product (

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...

) of a family of

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

A_i, i \in I

over a

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

''R'' is the associative algebra over ''R'' that is, roughly, defined by the generators and the relations of the

A_i

'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 In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...

of groups. In the category of commutative ''R''-algebras, the free product of two algebras (in that

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

) is their

tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...

Construction

We first define a free product of two algebras. Let ''A'' and ''B'' be algebras over a commutative ring ''R''. Consider their

tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...

, the direct sum of all possible finite tensor products of ''A'', ''B''; explicitly,

T = \bigoplus_^ T_n

where :

T_0 = R, \, T_1 = A \oplus B, \, T_2 = (A \otimes A) \oplus (A \otimes B) \oplus (B \otimes A) \oplus (B \otimes B), \, T_3 = \cdots, \dots

We then set :

A * B = T/I

where ''I'' is the two-sided ideal generated by elements of the form :

a \otimes a' - a a', \, b \otimes b' - bb', \, 1_A - 1_B.

We then verify the universal property of

holds for this (this is straightforward.) A finite free product is defined similarly.

References

*K. I. Beidar, W. S. Martindale and A. V. Mikhalev, ''Rings with generalized identities,'' Section 1.4. This reference was mentioned in

External links

* Abstract algebra {{algebra-stub