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So I'm new to tensor products and there's something that's been confusing me ...

Suppose that $A$ and $B$ are $R$-modules, I know that $A \otimes_R B$ is an abelian group;

and what I understood is that $a + a'\otimes b + b' = a \otimes b + a' \otimes b + a \otimes b' +a' \otimes b' $...

But does that mean that we can't write $ a\otimes b + a' \otimes b' \neq a + a'\otimes b + b'$?

I realize it seems like a stupid question; but...

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There are some brackets missing from what you write, which makes it confusing. But yes you are right (in general) that $a \otimes b + a' \otimes b' \neq (a+a') \otimes (b+b')$. If you are new to tensors you can think of expanding the right hand side as like multiplying out the brackets: you must have the $a \otimes b'$ and $a' \otimes b$ terms as well.

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Thanks! May I ask why you write in general? The case I'm dealing with is when $A$ and $B$ are $\mathbb{K}$-algebras ($\mathbb{K}$ is a field) and I am adding elements in $A \otimes_\mathbb{K} B$. Does that have any significance whatsoever? –  Kerry H Jan 23 at 8:02
    
You're welcome. I say "in general" because they would be equal if $a'=b'=0$ for example. –  mt_ Jan 23 at 10:21

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