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 (*xy* − *yx*). This defines a functor **Ring** → **CRing** 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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