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| emath... |
 Posted: Fri Nov 06, 2009 4:05 am |
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Guest
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Hi,
Are there any monoids that the group of units of it is isomorphic to
itself?
It is known that for a monoid M, the group of units of the Bruck-
reilly extension of M is isomorphic to the group of units of the
monoid M (i.e. U(BR(M,\theta))=U(M)). So can we find any other monoids
or extensions satisfies the above argument?
Thanks for help. |
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| Hagen... |
| Posted: Fri Nov 06, 2009 5:21 am |
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Guest
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[quote]Hi,
Are there any monoids that the group of units of it
is isomorphic to
itself?
It is known that for a monoid M, the group of units
of the Bruck-
reilly extension of M is isomorphic to the group of
units of the
monoid M (i.e. U(BR(M,\theta))=U(M)). So can we find
any other monoids
or extensions satisfies the above argument?
Thanks for help.
[/quote]
If I understand your question right, you are interested
in extensions M' of a monoid M such that M and M'
possess the same group of units.
Algebraic number theory provides us with a lot of
such monoids: the monoid M of non-zero integers
with multiplication as composition has the units -1
and +1.
Take a field extension K of degree 2 over the rationals
that is contained in the reals. Let R be the integral
closure of Z in K and let M' be the monoid R \ 0
again with multiplication as composition.
Then Dirichlet's theorem yields that the units of M'
are precisely the roots of unity contained in K, which
by assumption are only -1 and +1.
H |
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