The tensor product can be used as a means of taking of two subschemes in a scheme: consider the ${\displaystyle \mathbb {C} [x,y]}$-algebras ${\displaystyle \mathbb {C} [x,y]/f}$, ${\displaystyle \mathbb {C} [x,y]/g}$, then their tensor product is ${\displaystyle \mathbb {C} [x,y]/(f)\otimes _{\mathbb {C} [x,y]}\mathbb {C} [x,y]/(g)\cong \mathbb {C} [x,y]/(f,g)}$, which describes the intersection of the algebraic curves f = 0 and g = 0 in the affine plane over C.

intersections

More generally, if ${\displaystyle A}$ is a commutative ring and ${\displaystyle I,J\subseteq A}$ are ideals, then ${\displaystyle {\frac {A}{I}}\otimes _{A}{\frac {A}{J}}\cong {\frac {A}{I+J}}}$, with a unique isomorphism sending ${\displaystyle (a+I)\otimes (b+J)}$ to ${\displaystyle (ab+I+J)}$.

Tensor products can be used as a means of changing coefficients. For example, ${\displaystyle \mathbb {Z} [x,y]/(x^{3}+5x^{2}+x-1)\otimes _{\mathbb {Z} }\mathbb {Z} /5\cong \mathbb {Z} /5[x,y]/(x^{3}+x-1)}$ and ${\displaystyle \mathbb {Z} [x,y]/(f)\otimes _{\mathbb {Z} }\mathbb {C} \cong \mathbb {C} [x,y]/(f)}$.

Tensor products also can be used for taking of affine schemes over a field. For example, ${\displaystyle \mathbb {C} [x_{1},x_{2}]/(f(x))\otimes _{\mathbb {C} }\mathbb {C} [y_{1},y_{2}]/(g(y))}$ is isomorphic to the algebra ${\displaystyle \mathbb {C} [x_{1},x_{2},y_{1},y_{2}]/(f(x),g(y))}$ which corresponds to an affine surface in ${\displaystyle \mathbb {A} _{\mathbb {C} }^{4}}$ if f and g are not zero.

products

Given ${\displaystyle R}$-algebras ${\displaystyle A}$ and ${\displaystyle B}$ whose underlying rings are , the tensor product ${\displaystyle A\otimes _{R}B}$ becomes a graded commutative ring by defining ${\displaystyle (a\otimes b)(a'\otimes b')=(-1)^{|b||a'|}aa'\otimes bb'}$ for homogeneous ${\displaystyle a}$, ${\displaystyle a'}$, ${\displaystyle b}$, and ${\displaystyle b'}$.

graded-commutative rings

Extension of scalars

Tensor product of modules

Tensor product of fields

Linearly disjoint

Multilinear subspace learning

Kassel, Christian (1995), , Graduate texts in mathematics, vol. 155, Springer, ISBN 978-0-387-94370-1.

Quantum groups

Lang, Serge (2002) [first published in 1993]. Algebra. Graduate Texts in Mathematics. Vol. 21. Springer.  0-387-95385-X.

ISBN