Category of Rings - Subcategories - Category of Commutative Rings

The category of commutative rings, denoted CRing, is the full subcategory of Ring whose objects are all commutative rings. This category is one of the central objects of study in the subject of commutative algebra.

Any ring can be made commutative by taking the quotient by the ideal generated by all elements of the form (xyyx). This defines a functor RingCRing which is left adjoint to the inclusion functor, so that CRing is a reflective subcategory of Ring. The free commutative ring on a set of generators E is the polynomial ring Z whose variables are taken from E. This gives a left adjoint functor to the forgetful functor from CRing to Set.

CRing is limit-closed in Ring, which means that limits in CRing are the same as they are in Ring. Colimits, however, are generally different. They can be formed by taking the commutative quotient of colimits in Ring. The coproduct of two commutative rings is given by the tensor product of rings. Again, it's quite possible for the coproduct of two nontrivial commutative rings to be trivial.

The opposite category of CRing is equivalent to the category of affine schemes. The equivalence is given by the contravariant functor Spec which sends a commutative ring to its spectrum, an affine scheme.

Read more about this topic:  Category Of Rings, Subcategories

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