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 R→S. 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 R→S that admits an inverse ring homomorphism S→R 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 R→S.

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 φ is the set kerφ:={a∈R∣φ(a)=0}. It is easy to see that kerφ is an ideal of R. Conversely, any ideal I⊂R is the kernel of a suitable homomorphism R→S to be explained in the next subsection.

Furthermore, given an arbitrary homomorphism φ:R→S, one can move back and forth between ideals of S and of R: given an ideal J⊂S, the inverse image φ−1(J)⊂R is an ideal of R, while given an ideal I⊂R one obtains the smallest ideal of S containing the set φ(I). The first such move is called a contraction and the ideal φ−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.

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=∑i=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 R→R/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 R→R/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 R→R/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.

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....