These notes are based on a lecture on Commutative Algebra at the University of Luxembourg. A variety of sources including Eisenbud, the Princeton Companion to Mathematics and a Classic ‘Algebra’ text by MacLane and Birkhoff have been used to compile these notes. Hopefully it will be of benefit to those studying this course.

**1. Lecture Notes:Axioms for Rings **

A *ring* R = (R,+,,1) is a set R with two binary operations, addition and multiplication, and a nullary operation, ‘select 1’, such that

- (i) (R,+) is an abelian group under addition;
- (ii) (R,
,1) is a monoid under multiplication; - (iii) Multiplication is distributive (on both sides) over addition.

The last requirement means that all triples of elements a,b,c in R satisfy the identities

A commutative rign is one in which multiplication is commutative. Familiar systems of numbers are commutative rings under the usual operations of sum and product; examples are

Lets consider a rather trivial example of a ring, the set containing just the element 0, with addition and multiplicatio given (in the only possible way!) by 0 + 0, 00=0, is a ringl it will be called the ‘trivial ring’. \paragraph*{}The definition of a ring amounts to the statement that a ring is a set R with a selected element

for all

which containts the *unit* 1 and a *zero*0 such that

which cotaints to each element a an *additive inverse*(-a) with

and in which both distributive laws (1) hold. As an example of a non-commutative ring, we can consider the ring of square matrices. Every field – like for instance the real numbers and the complex numbers is an example of a commutative ring. **algebraically closed** every non-constant polynomial with coefficients in

** 1.1. Residue Class Rings **

: \left(\frac{\mathbb{Z}}{n\mathbb{Z}},+,\cdotp \right)\;\;\;n \in \mathbb{N}, n > 1 p \in \mathbb{P}\;\;prime, then (\frac{\mathbb{Z}}{p\mathbb{Z}},+,\cdotp) = \mathbb{F}_{p}\;\;is\;a\;field\;(finite) **Subrings**

As with other algebraic structures, there are several ways of forming new rings from old ones: for instance we can take subrings and direct products of two rings. Slightly less obviously, we can start with a ring R and form the ring of all polynomials with coefficients in R. We can also take QUOTIENTS, but in order to discuss these we must introduce the notion of an ideal.

** 1.2. Ideals **

A typical quotient construction for an algebraic structure A will identify some substructure B and regard two elements of A as ‘equivalent’ if they ‘differ by an element of B.’ If A is a group or a **vector space** then B will be a subgroup or a subspace. However the situation with rings is slightly different. To see why we need to think of quotients in another way. Lets consider them as images of homomorphisms. The substructures we would like to quotient by are the kernels of these homomorphisms, so we should ask ourselves what the kernel of a ring homomorphism (that is, the set of elements that map to 0) will be like. \paragraph*{} If *ideal.* For example, the set of all even integers is an ideal in

** 1.3. Modules **

Modules are to rings as vector spaces are to fields. In other words, they are algebraic structures where the basic operations are addition and scalar multiplication, but now the scalars are allowed to come from a ring rather than a field. For an example of a module over a ring that is not a field, take any Abelian group G. This can be turned into a module over *basis* of a module to be a linearly independent set of elements that span the module. However, many useful facts about bases in vector spaces do not hold for modules. For instance, in

**2. Varieties **

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in classical (and to some extent, modern) algebraic geometry. The word “variety” is employed in the sense of a mathematical manifold, for which, in Romance languages, cognates of the word “variety” are used. Historically, the fundamental theorem of algebra established a link between algebra and geometry by saying that a monic polynomial in one variable over the complex numbers is determined by the set of its roots, which can be considered a geometric object. Building on this result, Hilbert’s Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and subsets of affine space. Using the Nullstellensatz and related results, we are able to capture the geometric notion of a variety in algebraic terms as well as bring geometry to bear on questions of ring theory. Two simple examples of varieties are the circle and the parabola, which can be defined by the polynomial *algebraic set*, and it is called a *variety* if it cannoy be written as a union of smaller algebraic sets. However the concept is much more general: varieties can live in

**Affine Varieties**

**3. Irreducibles and factorization **

In this section, we examine the posibility of factorising elements of a ring into ‘irreducible’ elements (which cannot themselves be further factorized), and look at a special class of rings in which the analogue of the Fundamental Theorem of Arithmetic holds. \paragraph{}* First, we will make some simplifying assumptions about the ring R. We always assume that R is commutative, so that we regard ab and ba as essentially the same factorization of a ring element. (So, in a factorization, we do not care about the order of the factors.) Also, we exclude the divisors of zero. For, if ab = 0, then ac=a(b+c) for any element c, and there is little chance of unique factorizations. Accordingly, we assume, in this section and the next two, that R is an integral domain.

Definition 1Let R be an integral domain

- An element p
R is irreducibleif p is not zero or a unit, and if, whenever p =ab, either a or b is a unit (and the other is an associate of p).- R is a
unique factorization domainorUFDif it holds that (a) every element other than zero and units can be factorized into irreducibles; (b) if, where the and are irreducibles, then m = n, and (possibly after reordering and are associates for i = 1,…,n. Alternatively we can say that the ‘Fundamental Theorem of Arithmetic’ says that

is a UFD.

** 3.1. Zero-Divisors,Nilpotent Elements, Units **

A *zero-divisor* in a ring A is an element x which “divides 0”, i.e., for which there exists y

**4. The Nullstellensatz **

\cite{Atiyah68,Eisenbud} This is one of the first fundamental theorems of algebraic geometry. It controls the correspondence between affine algebraic sets and ideals; in particular, it enables us to calculate I(V(I)). We have already seen that certain problems arise when k is not algebraically closed. We therefore assume henceforth that:

**k is algebraically closed**

Definition 2In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical (this can be phrased as being a fixed point of an operation on ideals called ‘radicalization’).

Intuitively the Nullstellensatz allows one to create an algebra-geometry dictionary.

\newpage \appendix

**5. Summary of useful results from algebra **

** 5.1. Rings **

We denote by (x) or xA the idea generated by x in A, i.e. the set of all elements of the form xa with

- a.
*The isomorphism theorem.*Letbe a ring homomorphism and set I = Ker f. Let J be an ideal of J contained in I and let be the canonical projection. Then: - 1) There is a unique homomorphism
such that f= (we say that f factors through A/J), - 2)
is injective if and only if J = I, - 3)
is surjective if and only if f is. In particular, Im fA/Ker f. - b.
*Universal property of rings of polynomials.*Let A and B be two rings. Giving a homomorphism f:is equivalent to giving its restriction to A ( *i.e.,*a homomorphism from A to B) and the images of the variables( *i.e.,*n elements in B) - c.
*Euclidean division.*Let A be a ring and consider a polynomial P A[X],whose leading coefficient is a unit. For all there are Q,R such that F = PQ + R and or R = 0. For example, in k[X,Y] we can divide by with respect to Y, but not by XY – 1. - d.
*Products of rings.*The direct product of two rings A an B is the product set(i.e. , pairs(a,b) with product laws: (a,b) + (a’,b’) = (a+a’,b+b’),(a,b)(a’,b’)=(aa’,bb’)).

** 5.2. Ideals **

*Operations on ideals.*An arbitrary intersection of ideals is an ideal. The*sum*of a family of ideals of A is the set of*finite*sumswith . This is an ideal which is the upper bound of the ideals for inclusion. We denote it by. In particular, if =, then we obtain the *ideal generated*by the elementsIn the sum of the ideals generated by (x) and (y) is the ideal generated by the gcd of x and y.

The*product*of two ideals I and J is the ideal denoted by IJ*generated*by products xy with. We have IJ I J, but the converse is false: in the product ideal (resp. intersection) of (x) and (y) is the ideal generated by xy (resp. by the lcm of x and y).- b.
*Prime ideals.*An*integral domain*is a ringsuch that \paragraph{} For example, a field is an integral domain, a subring of an integral domain is an integral domain, and the ring of polynomials over an integral domain is an integral domain. \paragraph{} An ideal p of A is said to be *prime*if A/p is an integral domain. We note that inverse image of a prime ideal under a homomorphism is a prime ideal. \paragraph{} An ideal m is said to be maximal if it is maximal for inclusion amongst the ideals of A different from A. Equivalently, A/m is a field, called the*residue field*of m. It follows that any maximal ideal is prime. It can be proved using Zorn’s lemma that any ideal is contained in a maximal ideal. *Ideals of a quotient*Let A be a ring, I an ideal and p the canonical projection from A to A/I. The ideals of A/I are in (increasing) bijection with the ideals of A containing I via the mapsand Moreover, the prime ideals (resp. maximal ideals) correspond under this bijection. *Nilpotent elements.*An elementis said to be *nilpotent*if there is an integersuch that The set of nilpotent elements form an ideal called the *nilradical*of A. This ideal is the intersection of all the prime ideals of A. A ring without non-zero nilpotent elements is said to be*reduced*

** 5.3. Kernel: Ring Homomorphisms **

Let R and S be unital rings and let f be a ring homomorphism from R to S. If

Let us study the affine line

Definition 3(a) Let V be a closed set. V is called irreducible if for every decomposition V =with , closed we have or . (b) An algebraic set which is irreducible is called a variety.