A commutative ring is a set with two laws of composition, and obeying the following properties
is an abelian group with identity
is a commutative monoid
distribution, ,
A unit is an element of a ring with a multiplicative inverse.
A subring is a subset of a ring closed under the usual operations with the element .
Polynomials
A polynomial of with coefficients in is a finite linear combination of powers of the variable
The set of these form the polynomial ring
The monomials are independent, thus iff every coefficient is equal.
The degree of , is the largest such that the . A polynomial is monic if the leading coefficient is
Division is performed with a monic polynomial , where for all , there are unique such that where or
A ring with multiple variables is denoted
Ideals
A ring homomorphism is a map such that
If is an isomorphism, then are isomorphic
Substitution principle For , there is unique homomorphism that sends , and similarly for multivariate polynomial rings. Using this, we can also construct from .
An ideal of a ring is a nonempty subset of that is closed under addition, and ,
The unit ideal is or the ring itself. The zero ideal is . An ideal is proper if it not unit or zero. It is principal if it can be generated by a single element.
Quotients
Let be a ring homomorphism with kernel . Let be an ideal contained in , and the canonical map. Then there is a unique homomorphism such that . (First Isomorphism Theorem) Furthermore, if is surjective and , is an isomorphism.
Correspondence Theorem Let be a surjective ring homomorphism with kernel . There is a bijective correspondence between the ideals of and ideals of containing . goes to , and goes to .
Adjoining to yields the ring . This can be constructed by taking a polynomial with root , and considering .
Products
The product of rings is given by with operations defined
An idempotent is an element in a ring such that . Then is also an idempotent, where and . Moreover,
Fractions
Let be an integral domain. Consider fractions as elements of the form , where and . Then if , with operations defined and . Define the fraction field as the equivalence classes of fractions of elements of .
If is a subring of a field, embeds as a subfield.
Maximal Ideals
A maximal ideal of is some that is not equal to , but not contained in any other ideal.
A surjective ring homomorphism to a field has a maximal ideal for its kernel.
The maximal ideals of are where is a prime.
The maximal ideals of are principal ideals generated by monic irreducible polynomials.
Hilbert’s Nullstellensatz The maximal ideals of the polynomial ring correspond to points in , where correspond to kernel of map that sends , generated by polynomials .