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Free product of associative algebras
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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.
This article relies largely or entirely on a single source. (May 2024) |
In the category of commutative R-algebras, the free product of two algebras (in that category) is their tensor product.
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Construction
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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, where
We then set
where I is the two-sided ideal generated by elements of the form
It is then straightforward to verify that the above construction possesses the universal property of a coproduct.
A finite free product is defined similarly.
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References
- K. I. Beidar, W. S. Martindale and A. V. Mikhalev, Rings with generalized identities, Section 1.4. This reference was mentioned in "Coproduct in the category of (noncommutative) associative algebras". Stack Exchange. May 9, 2012.
External links
- "How to construct the coproduct of two (non-commutative) rings". Stack Exchange. January 3, 2014.
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