# Abstract Nonsense

## Functorial Properties of the Tensor Product (Pt. III)

Point of Post: This is a continuation of this post.

$\text{ }$

The important thing to note about the previous two theorems is that they really only require (for ingredients) left or right $R$-modules and $R$-maps, and the only parts of the proof, in fact the entire notion of exactness, really only has to do with the underlying abelian group structure. From this we can glean the following two theorems:

$\text{ }$

Theorem: If $M,N,L$ are right $(S,R)$-bimodules and we have an exact sequence

$\text{ }$

$M\xrightarrow{f}N\xrightarrow{g}L\to0$

$\text{ }$

is an exact sequence of $(S,R)$-maps, and if $A$ is a left $(R,T)$-module, then

$\text{ }$

$M\otimes_R A\xrightarrow{f\otimes 1_A}N\otimes_R A\xrightarrow{g\otimes 1_A}L\otimes_R A\to 0$

$\text{ }$

is an exact sequence of $(S,T)$-modules and $(S,T)$-maps.

$\text{ }$

and

$\text{ }$

Theorem: If $M,N,L$ are right $(R,T)$-bimodules and we have an exact sequence

$\text{ }$

$M\xrightarrow{f}N\xrightarrow{g}L\to0$

$\text{ }$

is an exact sequence of $(R,T)$-maps, and if $A$ is a left $(S,R)$-bimodule, then

$\text{ }$

$A\otimes_R M\xrightarrow{1_A\otimes f}A\otimes_R N\xrightarrow{1_A\otimes g}A\otimes_R L\to 0$

$\text{ }$

is an exact sequence of $(S,T)$-bimodules and $(S,T)$-maps.

$\text{ }$

$\text{ }$

References:

[1] Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.

[2] Rotman, Joseph J. Advanced Modern Algebra. Providence, RI: American Mathematical Society, 2010. Print.

[3] Blyth, T. S. Module Theory. Clarendon, 1990. Print.

[5] Grillet, Pierre A. Abstract Algebra. New York: Springer, 2007. Print.

January 4, 2012 -

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