A Refresher in Commutative Rings
31 Dec 2018In this post, we review the basic results from a first course in commutative ring theory.
Rings
The archetypal example of a ring is the set of integers , with the operations of addition and multiplication. Note that is an abelian group, while or even is not, since most integers do not have multiplicative inverses.
A commutative ring with identity (or simply ring) is an abelian group equipped with another binary operation (usually written as concatenation, ie. ), which is commutative, associative, and distributive over (ie. ), with identity element . An element of is called a unit if it has a multiplicative inverse.
An integral domain is a ring with no zero divisors, ie. if then or . A field is a ring in which every nonzero element is a unit.
Proposition: Every field is an integral domain.
Ideals
An ideal in a ring is an additive subgroup closed under -multiplication, ie.
is a proper ideal if .
For instance, the principal ideal generated by is defined by
is a maximal ideal of if there is no ideal with . By Zorn’s lemma, we can show that every proper ideal is contained in some maximal ideal.
is a prime ideal of if
The sum and product of two ideals are defined by
We can check that , , and are all ideals, which satisfy the inclusions
Quotient rings
It is easy to show that the (additive) quotient group is in fact also a ring (called the quotient ring), with operations given by
Proposition: Let be an ideal. Then:
- is maximal if and only if is a field;
- is prime if and only if is an integral domain.
Corollary: Every maximal ideal is prime.
Coprime ideals
Two ideals are coprime or relatively prime if . Note that if are coprime ideals then
so . Also, if is coprime to each of , then
so is coprime to . Hence by induction we have:
Proposition: If are pairwise coprime ideals, then .
For any two ideals , form the product of the two quotient maps to obtain a ring homomorphism . This has kernel , so we get the injective ring homomorphism
If are coprime, there exists , such that . Now for any , note that . Hence is surjective, thus so is , ie. is an isomorphism. By induction, we get:
Chinese Remainder Theorem: If are pairwise coprime ideals, then
Local rings
A local ring is a ring with a unique maximal ideal. We say that is a local ring if is a local ring with maximal ideal . In this case, the residue field of is , and we also say that is a local ring.
Note that if is a local ring, then every is not contained in any maximal ideal of , ie. is a unit. Conversely, if the set of non-units of form an ideal , then contains all proper ideals of , and hence is the unique maximal ideal of , so is a local ring.
Integral domains, PIDs, UFDs
Let be an integral domain throughout this section.
The most important elementary property of integral domains is the cancellation law:
for any with .
An element is prime if is a prime ideal. An element is irreducible if is not a unit, and for any factorisation , either or is a unit. Equivalently, is irreducible if is maximal among proper principal ideals.
Proposition: Every prime in is irreducible.
The standard counterexample for the converse is the element in the integral domain , which is irreducible (by considering the norm), but not prime (since but ).
The prime factorisation of an element, if it exists, is unique up to unit multiples and the order of factors:
Proposition: If has two factorisations into products of primes, say
then , and after some reordering of the we have for all .
Proof: prime implies that some , say . But is prime, hence irreducible, so is a unit. Then cancellation gives
so we replace by and finish by induction.
A principal ideal domain (PID) is an integral domain in which every ideal is principal. For instance, is a PID because every nonzero ideal is generated by its smallest positive element. Also, if is a field, then the polynomial ring is a PID, because every nonzero ideal is generated by a nonzero element of minimal degree.
Proposition: Every nonzero prime ideal in a PID is maximal.
Proposition: Every irreducible element of a PID is prime.
Proof: If is irreducible then is maximal among proper principal ideals, so is a maximal ideal, hence prime.
A unique factorisation domain (UFD) is an integral domain in which every nonzero, non-unit element has a prime factorisation (which must then be essentially unique, as we have shown).
Proposition: Every PID is a UFD.
Proof: Given a nonzero non-unit element , start with the trivial one-term factorisation and repeatedly split a term into a product of two non-unit elements. If this process terminates, we have a factorisation of into irreducibles, which are primes in a PID.
Otherwise, Kőnig’s lemma implies that every infinite binary tree has an infinite path, so there is a sequence of ideals
The union of these ideals is a new ideal, say ; but for some , so for all , contradiction.
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