By the references I gave before it would be my opinion that (at least) several (of the main) experts on the area believe the opposite to what you have stated. In any case, the statement (now) seems to be clear. Thanks for your opinion.

I was just using (what I suppose is) the standard convention (e.g. in the book of J. McConnell and J. Robson, or in the one of F. Anderson and K. Fuller): ideal = two-sided ideal. Thanks for the comment.

I meant $\bar{F}.\bar{F}$, $\bar{F}$ denoting the augmentation ideal. I've already edited it. Thanks for pointing it out.