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I have following question. I will explain it with abstract notation although in fact I am working with some peculiar matrices.

I have finitely presented noncommutative monoid with identity element [itex] I [/itex]. Presentation of this let say would be [itex] M = <S,T;S^2> [/itex] which means that if [itex] S,T [/itex] are generators of free monoid [itex] F=<S,T> [/itex] then [itex] S^2 =I [/itex] where [itex] I [/itex] is identity of monoid [itex] SI=TI=IT=IS[/itex]. Then [itex] M [/itex] is quotient of free monoid [itex] F [/itex] by the given relation [itex] M= F / [S^2-1] [/itex].

Now I have to construct ring [itex] R[M] [/itex] over rationals ( complex, whatever) with structure I will build by canonical method, as a sum of elements of monoid [itex] M [/itex] "multiplied" by weights from field R (or even C, or whatever). Here I found that general element [itex] Z [/itex] of ring [itex] R[M] [/itex] will be defined by expression:

[itex] Z = aI + bS +cT +dL [/itex]

where [itex]a,b,c,d[/itex]* are in the field [itex]R[/itex] where [itex] L [/itex] is certain element which is not an element of monoid [itex]M [/itex] but it is properly constructed element of [itex]R[M][/itex]! Namely [itex]L = [S,T] = ST - TS [/itex]. I point that monoid [itex]M[/itex] is multiplicative and noncommutative so [itex]L[/itex] is not present in monoid.

This is strange for me, and surprised me. I did not thought that it may happened: additional generator for a ring is required.

So I have situation, that monoid is generated by two generators [itex]<S,T>[/itex], while its ring over rationals [itex]R[M][/itex] by three [itex] <S,T,L>[/itex] ! In fact it is even finitely presented Lie algebra for which I have structure constants computed. Presently I am looking for its matrix representations different from starting one.

This is where my knowledge ends. I am looking for some bibliography in above matter. The only things I have found was about group rings and so, then I cannot qualify if different numbers of generators is something normal or strange? Typical or interesting? Did You ever see some books or papers with other but concrete examples of such objects ( monoid rings,algebras, modules over a field )? Maybe there are even some theorems in the wild and some of You knows where may I found them?

Best regards

Kazek

* - in above term [itex]aI[/itex] is not needed in fact, as I have relation [itex]S^2=I[/itex], but it has nice shape as it is, so in this post it does not matter.

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# Ring over monoid?

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