What does Hom(M,N) mean? Atiyah Macdonald proposition 2.9

Question

In Atiyah Macdonald, "Introduction to commutative Algebra" it says:

Proposition 2.9:i) Let $M' \xrightarrow[]{u}M \xrightarrow[]{v} M'' \rightarrow 0$ be a sequence of A-modules and homomorphisms. Then the sequence (4) is exact $\iff$ for all A-modules N, the sequence

$0\rightarrow Hom(M'',N)\xrightarrow[]{\bar{v}}Hom(M,N)\xrightarrow[]{\bar{u}} Hom(M',N)$ is exact.

Now my question: What does $Hom(M'',N)$ mean? I have looked in the text and on wikipedia and neither of them give me a straightforward answer. Part of me wants to say that it is a (group,category,module...something) which contains the homomorphisms from $M''$ to $N$.However if this was the case what would the maps $\bar{u}$ and $\bar{v}$ be?

Please note my knowledge of categories is not crash hot so please try and keep categorical comments as simple as possible.

Answer 1

$\operatorname{Hom}(M,N)$ refers to the set of $A$-module homomorphisms from $M$ to $N$. These form an abelian group under pointwise addition (define $f+g$ by $(f+g)(x)=f(x)+g(x)$), and if $A$ is commutative they in fact form an $A$-module by pointwise scalar multiplication (define $a\cdot f$ by $(af)(x)=a\cdot f(x)$).

Given an $A$-module homomorphism $g:M\to P$, there is an induced map $\bar{g}:\operatorname{Hom}(P,N)\to\operatorname{Hom}(M,N)$ that sends a homomorphism $f:P\to N$ to the composition $f\circ g:M\to N$. This map $\bar{g}$ is a homomorphism of abelian groups, and additionally a homomorphism of $A$-modules if $A$ is commutative.

Answer 2

You can find good information about the topic in Hungerford's book "Algebra", page 199 (specially theorem 4.1).

If $A$ and $B$ are modules over a ring $R$, then $Hom_R(A,B)$ is the set of all $R$-module homomorphisms $f:A\to B.$ The homomorhisms are expressed in the book.
Let A,B,C,D he modules over a ring R and $f:C\to A$ and $g:B\to D$, R-module homomorphisms. Then the map $\theta:Hom_R(A,B)\to Hom_R(C,D)$ given by $h\mapsto ghf$ is a homomorphism of abelian groups.

In your questions case:
$\bar{v}$ is composition of three homomorphism and maps a homomorphism, $f:M''\to N$ to a homomorphism $M\to N$ in this way:
$$M \xrightarrow[]{v} M'' \xrightarrow[]{f} N \xrightarrow[]{id_N}N.$$
Now for $\bar u$, ...

Answer 3

Both notations that bother you are defined and explained beginning at the top of page 18, right before the beginning of the subchapter "Submodules and quotient modules". Read carefully, don't skip paragraphs - the book is a classic written by two giants in that field, so you won't find serious omissions in it.