Take $S$ and $T$ two subsets of homogeneous elements of the associative noncommutative free algebra $F$ over a field $k$ generated by a set $X$, provided with a positive grading. To exclude pathologies $S$ and $T$ are contained in $\bar{F}.\bar{F}$, $\bar{F}$ denoting the augmentation ideal of $F$. If $S$ and $T$ are minimal generator sets for their respective generated ideals, is it the same true for the set $S.T$ of products of elements of $S$ and $T$?
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$\begingroup$ What do you mean by «to exclude pathologies»? The set $F\cdot F$ is just $F$. $\endgroup$– Mariano Suárez-ÁlvarezCommented Apr 8, 2013 at 20:55
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$\begingroup$ I meant $\bar{F}.\bar{F}$, $\bar{F}$ denoting the augmentation ideal. I've already edited it. Thanks for pointing it out. $\endgroup$– bruceCommented Apr 8, 2013 at 21:15
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1$\begingroup$ Left ideals? Right ideals? Two-sided ideals? $\endgroup$– Will SawinCommented Apr 8, 2013 at 22:41
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$\begingroup$ 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. $\endgroup$– bruceCommented Apr 9, 2013 at 2:54
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$\begingroup$ In my opinion one should specify when ideals are two-sided. $\endgroup$– Fernando MuroCommented Jun 22, 2013 at 19:46
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