Part I

1 Basic introductory theory

1.1 Commutative rings and ideals

The most fundamental objective of this book is a commutative ring having a multiplicative identity. Throughout the text, one refers to it simply as a ring.

A map between the underlying structures of two rings R and S is a ring homomorphism (or simply, a homomorphism) if it is compatible with the respective operations and, in addition, takes the multiplicative identity element of R to the one of S. Such a map is denoted by RS. As usual, if no confusion arises, one denotes the multiplicative identity of any ring by 1, even if there is more than one ring involved in the discussion. A ring homomorphism RS that admits an inverse ring homomorphism SR is called an isomorphism. As is easily seen, any bijective homomorphism is an isomorphism.

Often a ring homomorphism will simply be referred to as a map provided the context makes itself understood.

A subgroup of the additive group of a ring R is called a subring provided it is closed under the product operation of R and contains the multiplicative identity of R.

An element a?R is said to be a zero divisor if there exists b?R, b?0, such that ab=0; otherwise, a is called a nonzero divisor. In this book, a nonzero divisor will often be referred to as a regular element. A sort of extreme case of a zero divisor is a nilpotent element a, such that an=0 for some n=1.

One assumes a certain familiarity with these notions and their elementary manipulation.

A terminology that will appear very soon is that of an R-algebra to designate a ring S with a homomorphism RS.

1.1.1 Ideals, generators, residue classes

Alongside a ring, the most central object in commutative algebra is an ideal. The abstract notion of an ideal is due to R. Dedekind, as a culmination, one could say, of his long work in shaping up number theory.

A subset I?R is an ideal when it satisfies the following conditions:

(i)

I is a subgroup of the additive group of R.

(ii)

If b?I and a?R, then ba?I.

The second condition is what makes a distinction from the notion of a subring. Actually, the notion of an ideal is naturally deduced as a structure bestowed by a homomorphism. Namely, the kernel of a homomorphism f is the set kerf:={a?R│f(a)=0}. It is easy to see that kerf is an ideal of R. Conversely, any ideal I?R is the kernel of a suitable homomorphism RS to be explained in the next subsection.

Furthermore, given an arbitrary homomorphism f:RS, one can move back and forth between ideals of S and of R: given an ideal J?S, the inverse image f-1(J)?R is an ideal of R, while given an ideal I?R one obtains the smallest ideal of S containing the set f(I). The first such move is called a contraction and the ideal f-1(J)?R is the contracted ideal-a terminology that rigorously makes better sense when R?S; in the second move, the resulting ideal is called the extended ideal of I.

It is easy to produce ideals at will at least in a theoretical way. The procedure depends on the following elementary concept.

Definition 1.1.1.

Let I?R be an ideal. A subset S?I is named a set of generators of I if, equivalently:

• I is inclusion wise the smallest ideal of R containing S.

• I is the intersection of the family of all ideals of R containing S.

• Every element of I can be written in the form c1a1+?+cmam, for suitable elements a1,.,am?R and c1,.,cm?S.

Going the opposite direction, it is clear that an arbitrary subset S?R of a ring generates an ideal I?R. One uses the notation I=(S) to indicate this construction. In the case where S={c1,.,cm} is a finite set, the notational symbols I=(c1,.,cm) and I=c1R+?+cmR=Si=1mciR are used interchangeably.

Thus, the main question about ideals is not how one finds them, but how they function departing from these abstract properties.

One notes that the set {0} is an ideal; it is convenient to think of the empty set as being a set of generators of {0}. The next simplest kind of ideal is one generated by a single element of the ring-such ideals are called principal ideals and have an important role in the first steps of number theory and the elementary theory of divisors.

Let I?R be an ideal in a ring R. Inspired by the old theory of integer number congruences, Dedekind and followers arrived at a second important abstraction, namely the notion of the ring of residue classes with respect to I.

As a first step, like in classical number congruences, one introduces an equivalence relation on R by decreeing that two elements a1,a2?R are equivalent (or congruent) with respect to (or modulo) I if a1-a2?I. This originates the residue class set R/I whose elements are the congruence classes thus defined and installs by default the residue map RR/I. From elementary group theory, R/I acquires the structure of an Abelian group (the only possible such structure if one requires that the natural map RR/I becomes a group homomorphism).

In order to endow R/I with a ring structure, one invokes the characteristic property of ideals to define a product of classes and such that the group homomorphism RR/I becomes a ring homomorphism (there is only one way to produce this, an observation first made explicit by Krull in [111]). One calls R/I the residue class ring of R by I. An alternative is the quotient ring of R by I, as long as there is no confusion with the quotient operation of two ideals, to be introduced in the next subsection.

One needs a notation for the residue class of an element a?R or, equivalently, for the image of a?R by the residue map. Rigorously, one should use a+I, but unfortunately this becomes increasingly cumbersome as calculations evolve. Therefore, it is usual to put a bar over the element- a? as it is-provided the ideal I is clear from the context.

One reason to consider these generalized congruences can be formulated in the following elementary result.

Proposition 1.1.2.

Let R?S be a surjective homomorphism of rings. Then there is an ideal I?R such that S?R/I and, moreover, this establishes a bijection between the set of surjective ring homomorphisms with source R, up to isomorphisms of the target, and the set of ideals of R.

The proof is left as a recap exercise; as in this book, one assumes familiarity with the so-called theorems of homomorphism (usually listed as first, second, etc.). These theorems were first proved by R....