Let $K$ be a field and $$ A=\left\{ \begin{pmatrix} a&b&c\\ d&e&f\\ 0&0&g \end{pmatrix} :a,\dots,g\in K \right\}, $$ then $$ J=\left\{ \begin{pmatrix} 0&0&c\\ 0&0&f\\ 0&0&0 \end{pmatrix} :c,f\in K \right\} $$ is a two-sided ideal of $A$. I want to show that J is projective as a left A-module.
Now let $f\colon M\to N$ be a surjective homomorphism of left A-modules and $g\colon J\to N$ be a homomorphism of left A-modules. Since $J=AT$, where $$ T= \begin{pmatrix} 0&0&1\\ 0&0&0\\ 0&0&0 \end{pmatrix}, $$ if I could define $h\colon J\to M$ by $h(X)=Ym$ when $X=YT\;(Y\in A)$ and $f(m)=g(T)$, then $fh=g$. But I'm not sure if $h$ is well-defined.
Hint: Your ideal is isomorphic (as a left module) to $Ae$ with $e=E_{1,1}$ the elementary matrix which is all zeroes except for a $1$ at the upper left corner. Set $f=1-e$, and consider the two left ideals $Ae$ and $Ef$: show $A=Af\oplus Ae$. it follows immediately that $Ae$ is projective.