Talk:Center (algebra)
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Subring
[edit]I don't think that one should write "The center is a commutative subring of R, so R is an algebra over its center."
This gives the impression that every ring R is an algebra over every commutative subring, which is false, take for example the quaternions as a ring and the complex numbers as a commutative subring. Maybe one should omit the word "so" and everything's fine? -- 130.83.2.27 10:51, 7 August 2007 (UTC)
- I've changed "so" to "and". Thanks for noticing.--Roentgenium111 (talk) 23:54, 3 March 2010 (UTC)
Functoriality
[edit]In the cases of groups and rings, the center defines a functor Grp → Ab and Ring → CRing. I think that merits each of those having their own page, or at least vastly expanded sections rather than disambiguation.
I don't know if in either case it is a left or right adjoint. I know the inclusion CRing → Ring is continuous, preserving products, pullbacks, the terminal object, and other limits, and I'd thought maybe the center of rings functor preserved colimits, but I'm not 100% sure that the coproduct of rings, the free product, is taken to the coproduct of commutative rings (the tensor product over ℤ). Does anyone know any more about this? I'd sure appreciate some references for further reading on the category of rings and commutative rings. Much thanks. --Daviddwd (talk) 21:45, 27 August 2014 (UTC)
- The center is not functorial, neither for groups nor for rings. The inclusion of {e, (1 2)} into S3, for example, is a group homomorphism that does not restrict to a homomorphism between centers. Also, the inclusion of C into H (the Hamiltonian quaternions) is a ring homomorphism out of a commutative ring whose image is not central, which means that H is not a C-algebra, but is an R-algebra because Z(H) = R. The inclusion functors do not preserve colimits, so they cannot have right adjoints. The left adjoints are given by abelianization and an analogous quotient for rings (quotient out by the 2-sided ideal generated by all ring-theoretic commutators xy − yx), respectively. GeoffreyT2000 (talk, contribs) 17:54, 21 July 2017 (UTC)