## Extension of Scalars (Pt. I)

**Point of Post: **In this post we discuss some of the fundamental ideas concerning extension of scalars.

*Motivation*

We finally can now get around to discussing a use of tensor products that I touted in my original introduction: extension of scalars. Namely, the idea is that we are handed some -module and some superring and would like to see to what extent we can consider to be an -module. In other words, are are considering the problem which is dual to taking an -module and considering it as an -module by just “restricting” scalars. Unfortunately, this is often impossible to do. For example, can’t be made into a -space since, if it could, whatever would be (and note, it has to be an INTEGER since the multiplication is a map ) it would satisfy which is impossible! So, the next best thing one could hope to do is perhaps extend the module in some “minimal” way so that it can be naturally imbued with an -module structure. In other words, we want to find some -module for which embeds into as an -module, and doing this in some minimal sort of way. For example, while cannot be given the structure of an -space it can surely be -embedded into such a space, namely itself. Once again, this may not always be possible, for example if is a finite abelian group (e.g. -module) then can never be -embedded into a -space since (as can be easily proven!) every element of a -space has infinite order. Thus, we can really only hope to ask for a “best case scenario”. What -module maximizes both the ability to faithfully (to some degree) embed and is minimal in some sense. In the case of our abelian group it’s clear that we’re going to have to take to be our -space since this is the only such space in which can be “embedded” (albeit very unfaithfully). This is what we mean by extension of scalars, such an -module .

If the obvious intellectual curiosity isn’t enough to motivate this problem I can mention that it has many uses. For example, I have in the past discussed the notion of induced representations which can be seen as extension of scalars problem. Namely, we suppose that we have some group and some subgroup . Roughly then what we wish to do is pass from an -representation to a -representation, which can be thought of as extending an -module (where is the group algebra) to an -module.

So, why might we expect that the tensor product is the correct route for such an extension of scalars? There is actually a quite natural way one might realize this. The first is the naive attempt that one might actually try to make a given -module into an -module in the most brutish way. Namely, let’s define a “formal multiplication” of and elements. Namely, given and let just be formal symbol, our “multiplication”. We then see that if this “multiplication” is to create a valid -module structure extending that of ‘s preexisting -module structure, we’re going to need certain identities to hold. For example, by mere definition of a module we are going to need that is linear in each entry (this is because we should have that , etc.). Moreover, since we want (since we are extending the -module structure) and we see that we are going to have for all , , and . Thus, we see that is an -biadditive map . Therefore if we’d like to consider a “universal” way to define an -module structure on it seems that we should be looking for a “universal” -biadditive map and so really we want to just be and that this should be our extension of scalars.

*Extensions of Scalars*

Let be some unital ring and . If is some left -module, we define the *extension of sclars from to *on to be the left -module where we are thinking of as an -bimodule in the usual way.

Note that we have an -module map given by . This is, indeed, an -map since

The reason then we can really justify rigorously that is the solution to the “extension of scalars problem” as outlined above:

**Theorem: ***Let be a unital subring of , and a left -module. Then, if is the above-described -map then is the largest quotient of -embeddable into an -module. In particular, is -embeddable into and -module and if is another such quotient then .*

**Proof: **The fact that can be -embedded into an -module is clear by the first isomorphism theorem. Suppose now that is some left -module and we have some -map . We then consider the map given by and note that it is clearly linear in each entry and for all so that is actually -biadditive. By the universal property of tensor products we get an -map such that . Thus, we see that if then so that so that . Thus, as desired.

In particular, we see that is -embeddable into an -module if and only if is trivial.

**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.

[4] Lang, Serge. *Algebra*. Reading, MA: Addison-Wesley Pub., 1965. Print.

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

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