10 Oct '07 13:40>1 edit
Originally posted by PalynkaSince the a[ i] aren't defined (we don't know if they are numbers, matrices, differential operators, integral operators or whatever) we can think like this:
But don't you have to multiply by 1/a[k] to get there, which is not guaranteed to be in the set?
a[1]*...*a[k]*...*a[n]=a[k] now if * is commutative between them, that was what I implicitly admitted before, we can write a[1]*...*a[k-1]*a[k+1]*...*a[n]*a[k]=a[k]. Let us define A=a[1]*...*a[k-1]*a[k+1]*...*a[n].
Thus we have A.a[k]=a[k] for every a[k] and since we have made no special hypothesis about the nature of a[k] it follows that A=1. Even if a[k]=0 A=1 is still valid. If we were talking about a particular a[k] then a[k] must be different from 0 like you say, but the thing is that we are talking about every a[k] so we never multiply by 1/a[k] we look at equation and come to the conclusion that A=1 is the only way out.
Another thing is that A=1 doesn't mean that A is the number 1 it should be read as "A is the unit operator".
Edit: Just seen your edit. 🙂