Commutative Rings and Fields — Studies in Algebra and Group Theory

N17 Commutative Rings and Fields

Definition N17.1 (Commutative rings). A commutative ring is a set with two operations, and , such that:

is an abelian group with identity ;
is a commutative semigroup with identity :
- for all ,
- for all ,
- for all ;
for all (distributive law).

Definition N17.2 (Fields). A field is a commutative ring such that for all there exists with . So a commutative ring, but with inverses, such that is an abelian group, not just a semigroup.

Remark (Remarks (on rings and fields)).

Ring axioms ⟹ cancellation law (for ).
The trivial ring is such that , and it is the only case where this is true (it is NOT a field).
In a field, or . This doesn’t need to be true in a ring, though.
In a field ⟹ cancellation laws for AND .

Definition N17.3 (Ring/field homomorphisms). A ring or field homomorphism is a map that respects the operations: (Because : when , then .)

Definition N17.4 (Ring of polynomials). Given a field , the “RoP” (ring of polynomials) in one formal variable is