Injectivity of Total Ring of Quotients.

Question

It is well known that $\mathbb{Q}$ is an injective $\mathbb{Z}$-module, and more generally if $R$ is a domain, then its field of fractions $\mathrm{Frac}(R)$ is an injective $R$-module. Now my question: Let $R$ be a commutative ring with unit and let $Q(R)$ be its total ring of quotients.

Is $Q(R)$ an injective $R$-module?

I think it is not, but I don't have a counterexample.

Answer 1

Not necessarily. Let $R={\Bbb Q}[x,y]/(x^2,xy,y^2)$. Then every element of $R$ is either a zero-divisor or invertible, so $R=Q(R)$. However $R$ is not injective as the $R$-module homomorphism from the ideal ${\Bbb Q}x+{\Bbb Q}y$ to $R$ which sends $x$ to $y$ and $y$ to $x$ cannot be extended to $R$.