TITLE: Is $(x)\otimes_{k[x]/(x^2)}(x)$ zero?
QUESTION [7 upvotes]: I am trying to decide if $(x)\otimes_{k[x]/(x^2)}(x)$ is zero. 

So I considered $x \otimes x$ which I rewrote as $1 \otimes x^2 = 1 \otimes 0 = 0$. But then I realized that $1$ does not live in $(x)$. Now I am unsure how to proceed. Also it's beginning to feel like $x \otimes x \neq 0$. Thanks.

REPLY [5 votes]: The following observations are made en passant in the accepted answer. I think they deserve a full proof.

Let $R=K[X]/(X^2)$, and $x$ be the residue class of $X$ modulo the ideal $(X^2)$. Then $$\operatorname{Ann}_R(x)=Rx.$$ 

Since $x^2=0$ we have $Rx\subseteq\operatorname{Ann}_R(x)$. For the converse, let $f(x)\in\operatorname{Ann}_R(x)$. Then $f(x)x=0\Rightarrow f(X)X\in(X^2)\Rightarrow f(X)\in(X)\Rightarrow f(x)\in Rx$.
Since $Rx\simeq R/\operatorname{Ann}_R(x)$ we get $Rx\simeq R/Rx$, and this gives us $$Rx\otimes_RRx\simeq R/Rx\otimes_RR/Rx\simeq R/Rx.$$