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The primary reference I'm using for basic algebra is Mac Lane/Birchoff's Algebra, 3rd Edition. In this text, on page 175, a free module is defined as follows:

Let $X$ be a subset of an $R$-module $F$ and $i:X \longrightarrow F$ the canonical injection of $X$ into $F$. Then, $R$ is called a free module on $X$ when to every function $f:X\longrightarrow A$ to an $R$-module $A$ there is exactly one linear map $t:F\longrightarrow A$ such that $t \circ i = f$

Now, I have encountered another definition frequently that doesn't appear to be equivalent It is:

A free module $F$ on a set $X$ is an $R$-module $F$ and a map $i:X\longrightarrow F$ such that for any $R$-module $A$ and any map $f:X\longrightarrow A$ there is a unique linear map $t:F \longrightarrow A$ such that $t \circ i = f$

This particular definition comes from page 151 of Paul Garrett's notes on abstract algebra available from his website. The obvious difference is that in Garrett's version, the set $X$ is not required to be a subset of $F$ and the function $i$ is not the inclusion but, seemingly, any function.

So, my questions:

  1. Which is the preferred/more common definition?

  2. I believe that Garrett's definition includes M&B's definition as a special case; is this so?

  3. Other than generality, is there any advantage of Garrett's definition over M&B's?

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In the second definition, you can identify $X$ with its image under $i$. There's no important difference between these two formulations. If $i$ were not an injection, then you wouldn't be able to apply the universal property to an injective $f$. –  Dylan Moreland Jan 13 '12 at 3:02
    
Consider the set-theoretic map $f:X\to R$ sending some fixed $x\in X$ to $1$ and all other elements $y\in X$ to $0$. Let $\bar f:F\to R$ be Garrett's extension of $f$ to $F$. Since $1=f(x)=\bar f(i(x))\neq0=f(y)=\bar f(i(y)) $, you see that $i(x)\neq i(y)$ and $i$ is thus necessarily injective. –  Georges Elencwajg Jan 13 '12 at 10:20

2 Answers 2

up vote 3 down vote accepted

Both definitions are saying essentially the same thing. I think Paul Garrett's definition is more elegant and more accepted today. Garrett's definition of the free module on $X$ is unique up to a unique isomorphism, while MacLane / Birchoff's definition of the free module on $X$ is unique, but only defined for subsets $X$ of the free module in question. Strictly speaking if you are given a set $X$ and want to form the free module on it, you can only do so with Garrett's definition. But everybody allows that MacLane/Birchoff intend for you to be able to do so as well, where you just identify the set $X$ with a basis of the free module $F$ that you construct. No doubt, they understood this distinction very well, and gave their definition in the hope of simplifying things for students. Now that category theory (pioneered by MacLane) is more accepted, there is less need for this simplification.

By the way, you should be able to prove that, with Garrett's definition, $i: X \rightarrow F$ is always injective as a map of sets.

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The two definitions are equivalent. To see this, not that the if $i$ is not an inclusion, we will not be able to factor all maps $f:X\to A$ through $i$, and that by virtue of $i$ being an inclusion we get $X\subseteq A$.

Edit: As Dylan said, there is no reason to distinguish between $X$ and its image under $i$.

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