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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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What do you mean by «to exclude pathologies»? The set $F\cdot F$ is just $F$. –  Mariano Suárez-Alvarez Apr 8 '13 at 20:55
    
I meant $\bar{F}.\bar{F}$, $\bar{F}$ denoting the augmentation ideal. I've already edited it. Thanks for pointing it out. –  bruce Apr 8 '13 at 21:15
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Left ideals? Right ideals? Two-sided ideals? –  Will Sawin Apr 8 '13 at 22:41
    
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. –  bruce Apr 9 '13 at 2:54
    
In my opinion one should specify when ideals are two-sided. –  Fernando Muro Jun 22 '13 at 19:46
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