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- Tensor_product_of_algebras abstract "In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras.Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, we may form their tensor productwhich is also an R-module. We can give the tensor product the structure of an algebra by definingand then extending by linearity to all of A ⊗R B. This product is easily seen to be R-bilinear, associative, and unital with an identity element given by 1A ⊗ 1B, where 1A and 1B are the identities of A and B. If A and B are both commutative then the tensor product is as well.The tensor product turns the category of all R-algebras into a symmetric monoidal category. There are natural homomorphisms of A and B to A ⊗R B given byThese maps make the tensor product a coproduct in the category of commutative R-algebras. (The tensor product is not the coproduct in the category of all R-algebras. There the coproduct is given by a more general free product of algebras).The tensor product of algebras is of constant use in algebraic geometry: working in the opposite category to that of commutative R-algebras, it provides pullbacks of affine schemes, otherwise known as fiber products.".
- Tensor_product_of_algebras wikiPageID "372395".
- Tensor_product_of_algebras wikiPageRevisionID "556247991".
- Tensor_product_of_algebras hasPhotoCollection Tensor_product_of_algebras.
- Tensor_product_of_algebras subject Category:Algebras.
- Tensor_product_of_algebras subject Category:Commutative_algebra.
- Tensor_product_of_algebras subject Category:Multilinear_algebra.
- Tensor_product_of_algebras subject Category:Ring_theory.
- Tensor_product_of_algebras type Abstraction100002137.
- Tensor_product_of_algebras type Algebra106012726.
- Tensor_product_of_algebras type Algebras.
- Tensor_product_of_algebras type Cognition100023271.
- Tensor_product_of_algebras type Content105809192.
- Tensor_product_of_algebras type Discipline105996646.
- Tensor_product_of_algebras type KnowledgeDomain105999266.
- Tensor_product_of_algebras type Mathematics106000644.
- Tensor_product_of_algebras type PsychologicalFeature100023100.
- Tensor_product_of_algebras type PureMathematics106003682.
- Tensor_product_of_algebras type Science105999797.
- Tensor_product_of_algebras comment "In mathematics, the tensor product of two R-algebras is also an R-algebra. This gives us a tensor product of algebras. The special case R = Z gives us a tensor product of rings, since rings may be regarded as Z-algebras.Let R be a commutative ring and let A and B be R-algebras. Since A and B may both be regarded as R-modules, we may form their tensor productwhich is also an R-module.".
- Tensor_product_of_algebras label "Tensor product of algebras".
- Tensor_product_of_algebras sameAs m.020m26.
- Tensor_product_of_algebras sameAs Q7700713.
- Tensor_product_of_algebras sameAs Q7700713.
- Tensor_product_of_algebras sameAs Tensor_product_of_algebras.
- Tensor_product_of_algebras wasDerivedFrom Tensor_product_of_algebras?oldid=556247991.
- Tensor_product_of_algebras isPrimaryTopicOf Tensor_product_of_algebras.