Let $R$ be a commutative ring with $1$ and let $A$ be a commutative $R$-algebra. We view $A\otimes A$ also as $R$-algebra, the multiplication on generatrs being given by $(a\otimes b)(a'\otimes b')= aa'\otimes bb'$.
Denote by $m: A\otimes A\rightarrow A$ the product homomorphism, i.e. the map that maps a generator $a\otimes b$ to $ab$.
My question is: why is the kernel of $m$ generated by the elements $a\otimes 1 - 1\otimes a$ where $a$ runs through $A$?
The answer seems to be so easy that I could not find it anywhere in books. Alas, it is not clear to me :(
Thank you for your help!