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bilinear map (Definition)

Let $R$ be a ring, and let $M$, $N$ and $P$ be modules over $R$. A function $f\colon M\times N\to P$ is said to be a bilinear map if for each $b\in M$ the function $h\colon N\to P$ defined by $h(y)=f(b,y)$ for all $y\in N$ is linear (that is, an $R$-module homomorphism), and for each $c\in N$ the function $g\colon M\to P$ defined by $g(x)=f(x,c)$ for all $x\in M$ is linear. Sometimes we may say that the function is $R$-bilinear, in order to make the base ring clear.

A common case is a bilinear map $V\times V\to V$, where $V$ is a vector space over a field $K$; the vector space with this operation then forms an algebra over $K$.

If $R$ is a commutative ring, then every $R$-bilinear map $M\times N\to P$ corresponds in a natural way to a linear map $M\otimes N\to P$, where $M\otimes N$ is the tensor product of $M$ and $N$ (over $R$).



"bilinear map" is owned by yark.
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See Also: multi-linear, bilinear form, scalar map

Other names:  bilinear function, bilinear operation, bilinear mapping, bilinear operator, bilinear pairing, pairing
Also defines:  bilinear

Attachments:
bilinearity and commutative rings (Theorem) by Algeboy
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Cross-references: tensor product, linear map, map, commutative ring, algebra, field, vector space, function, modules, ring
There are 43 references to this entry.

This is version 8 of bilinear map, born on 2005-12-01, modified 2007-01-14.
Object id is 7510, canonical name is BilinearMap.
Accessed 7882 times total.

Classification:
AMS MSC: 13C99 (Commutative rings and algebras :: Theory of modules and ideals :: Miscellaneous)
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