I thought a ring was commutative for another reason but I realized that something I had not yet discovered, had led me to look for the solution in the wrong place.
I see that 'commutative' property of a ring is when ring is 'center structured' or exist an central algebra but.. here for me is not clear because I don't understand from where or how emerge this subring and what is really this 'subring nature': a structure action for divison rings ? How it works ?
the center of a ring R is the subring consisting of the elements $x$ such that $xy$ = $yx$ for all elements y in $R$. It is a commutative ring and is denoted as $Z(R)$
But.. in other words a division ring is not-noncommutative and not-commutative when
Every division ring is therefore a division algebra over its center
but is like to say that a division ring is not noncommutative or commutative for definition but could reflect itself commutative or noncommutative but this is not yet a definite state, it is only a possibility that this can happen, but it has not happened yet. But mathematically this is fundamental because in this gap we can choose any morphism, any functor, any structure
In definition they do not say when this happens - in this way of exposing you reflect the lack or loss of the object that I need to see instead!
They already say when this is possible, but so they do not explain the 'how', therefore, the definition does not allow you to locate the right structure to have a field if you leave the ring.
if division ring is commutative same division ring is 'finite' and so is a field.
But if is this is true
- a subring is a 'subring' when exist or when we replace an algebra X to use the central algebra for a division ring to give to the ring a center ??
In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;2 every division ring arises in this fashion from some simple module
So.. problem is located in the selection of which endomorphism ring I choose to generate a 'commutivity' definition. Schur's lemma adds something (maybe some coefficients, look here how) that does not serve me or that I should remove because commutitive property can reflects at same times different ways or options to make the ring commutable.