Another possible answer, following Qiaochu's comment: The elements $x^2$ and $x^3$ (Edit: $x^5$ and $x^6$) in $k[x^2, x^3]$ (alternatively, $k[x,y]/(x^3-y^2)$), where $k$ is a nonzero ring.
Another possible answer, following Qiaochu's comment: The elements $x^2$ and $x^3$ in $k[x^2, x^3]$ (alternatively, $k[x,y]/(x^3-y^2)$), where $k$ is a nonzero ring.