Definition. A subset $I\subset k[x_1,\ldots,x_n]$ is an ideal if
i. $0\in I.$
ii. If $f,g\in I$, then $f+g\in I$.
iii. If $f\in I$ and $h\in k[x_1,\ldots,x_n]$, then $hf\in I$.
I think the point $i$ above implies directly that the multiplicative inverse is not defined in ideal. I want to make sure I understand this totally so is multiplicative inverse defined for ideal? No because $0\in I$? And $x^3 y\not\in \langle x^3y^3\rangle$?