A question about direct sums and products of modules

Question

I'm reading about tensor products of modules and I don't understand the following paragraph on page 2:

Even though elements of $M \times N$ and $M \oplus N$ are written in the same way, as pairs $(m, n)$, bilinear functions $M \times N \rightarrow P$ should not be confused with linear functions $M \oplus N \rightarrow P $. For example, addition as a function $ R \oplus R \rightarrow R $ is linear, but as a function $R \times R \rightarrow R$ it is not bilinear. Multiplication as a function $R \times R \rightarrow R$ is bilinear, but as a function $R \oplus R \rightarrow R$ it is not linear. Linear functions are generalized additions and bilinear functions are generalized multiplications.

My questions are:

1. Why the distinction between $\times$ and $\oplus$? Aren't they the same for finite sums (products)?

2. I don't understand how addition can be linear but not bilinear. Surely, if $(a,b) \mapsto a + b$ then it doesn't matter in which argument it is linear. (Thinking of e.g. $\mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}$)

3. And if $(a,b) \mapsto ab$ then how can it be linear in both arguments (bilinear) but not be in one (linear). It's difficult to write this sentence as it makes no sense to me whatsoever.

Many thanks for your help!

Answer 1

Bi-linear means that fixing one argument, the resulting function of the other argument is linear. In your example $(a,b)\mapsto a+b$ (say, over $\mathbb{Z}$), fixing $a =1$, the function $b\mapsto b+1$ is not linear since $b1 + b2\mapsto b1 + b2 +1$ and not to $(b1+1)+(b2+1)=b1+b2+2$. On the other, hand the function $(a,b)\mapsto a\cdot b$ is bilinear. for example, fixing $a = 2$ we get the function $b\mapsto 2b$ which is clearly linear in $b$. I guess that once you clear this up for yourself, the rest will follow easily.

EDIT: finite sums and finite products of modules are "the same", though this should be thought of as a "theorem" and not a definition. The correct way to address those issues is using the language of category theory and universal properties. To make a long story short, I will just say that linear functions from some module $L$ to $M\times N$ are in natural correspondence to pairs of linear functions, one from $L$ to $M$ and one from $L$ to $N$ while pairs of linear functions one from $M$ to $L$ and one from $N$ to $L$ are in natural correspondence with linear functions from $N\oplus M$ to $L$. While it is easy to see that this is "the same" for every pair of modules $M,N$ this breaks for infinite families of modules (try to find the corresponding modules and see why they are different). You can also think of those notions in other "categories" like groups or topological spaces (with group homomorphisms and continues functions resp.) and see that in general the "sum" and "product" are very different. If you are interested in learning the long story (which is very interesting in my opinion), you can try reading about categorical sum and categorical product on wikipedia that will also lead you naturally to the notion of universal property.