Is the following true?


Let $R$ be a commutative ring and $S$ be an $R$-algebra. Let $\mu: S\otimes\_R S\to S$ be the multiplication.
Let $I$ be the kernel of $\mu$, and $J\subseteq I$ be the smallest ideal containing $1\otimes s-s\otimes 1$ for all $s\in S.$


Then $J=I.$



Generators of the kernel of multiplication of a ring
Assuming by $J \subset I$ you mean $J\subset S\otimes\_R S$, yes. Let $ab=cd$. Then we must show that $a\otimes b-c\otimes d\in J$. But 


$a\otimes b-c\otimes d=a\otimes b+ab\otimes 1 - c\otimes d -cd\otimes 1$


$=a \otimes 1(1\otimes b-b \otimes 1) -c \otimes 1 (1 \otimes d -d \otimes 1)\in J$


An identical argument works for a longer formula like $a\_1b\_1+a\_2b\_2+\dots a\_nb\_n=c\_1d\_1+\dots+c\_md\_m$.


