Английская Википедия:Free product of associative algebras

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In algebra, the free product (coproduct) of a family of associative algebras <math>A_i, i \in I</math> over a commutative ring R is the associative algebra over R that is, roughly, defined by the generators and the relations of the <math>A_i</math>'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.

In the category of commutative R-algebras, the free product of two algebras (in that category) is their tensor product.

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, the direct sum of all possible finite tensor products of A, B; explicitly, <math>T = \bigoplus_{n=0}^{\infty} T_n</math> where

<math>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</math>

We then set

<math>A * B = T/I</math>

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

<math>a \otimes a' - a a', \, b \otimes b' - bb', \, 1_A - 1_B.</math>

We then verify the universal property of coproduct 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 Шаблон:Cite web

External links

• Шаблон:Cite web

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