Contents 1 Foundations 3 1 Commutative rings and their ideals . 3 1.1 Rings . 3 1.2 The category of rings . 4 1.3 Ideals . 6 1.4 Operations on ideals . 7 1.5 Quotient rings . 8 1.6 Zerodivisors . 9 2 Further examples . 9 2.1 Rings of holomorphic functions . 9 2.2 Ideals and varieties . 10 3 Modules over a commutative ring . 11 3.1 Definitions . 11 3.2 The categorical structure on modules . 12 3.3 Exactness . 14 3.4 Split exact sequences . 16 3.5 The five lemma . 17 4 Ideals . 18 4.1 Prime and maximal ideals . 18 4.2 Fields and integral domains . 19 4.3 Prime avoidance . 20 4.4 The Chinese remainder theorem . 21 5 Some special classes of domains . 22 5.1 Principal ideal domains . 22 5.2 Unique factorization domains . 22 5.3 Euclidean domains . 23 6 Basic properties of modules . 24 6.1 Free modules . 24 6.2 Finitely generated modules . 26 6.3 Finitely presented modules . 27 6.4 Modules of finite length . 29 Copyright 2011 the CRing Project. This file is part of the CRing Project, which is released under the GNU Free Documentation License, Version 1.2. 1 CRing Project, Chapter 1 2 Chapter 1 Foundations The present foundational chapter will introduce the notion of a ring and, next, that of a module over a ring. These notions will be the focus of the present book. Most of the chapter will be definitions. We begin with a few historical remarks. Fermat's last theorem states that the equation xn + yn = zn has no nontrivial solutions in the integers, for n ≥ 3. We could try to prove this by factoring the expression on the left hand side. We can write (x + y)(x + ζy)(x + ζ2y) ::: (x + ζn−1y) = zn; where ζ is a primitive nth root of unity. Unfortunately, the factors lie in Z[ζ], not the integers Z. Though Z[ζ] is still a ring where we have notions of primes and factorization, just as in Z, we will see that prime factorization is not always unique in Z[ζ]. (If it were always unique, then we could at least one important case of Fermat's last theorem rather easily; see the introductory chapter of [Was97] for an argument.) p p For instance, consider the ring Z[ −5] of complex numbers of the form a + b −5, where a; b 2 Z. Then we have the two factorizations p p 6 = 2 · 3 = (1 + −5)(1 − −5): Both of these are factorizations of 6 into irreducible factors, but they are fundamentally different. In part, commutative algebra grew out of the need to understand this failure of unique factor- ization more generally. We shall have more to say on factorization in the future, but here we just focus on the formalism. The basic definition for studying this problem is that of a ring, which we now introduce. x1 Commutative rings and their ideals 1.1 Rings We shall mostly just work with commutative rings in this book, and consequently will just say \ring" for one such. Definition 1.1 A commutative ring is a set R with an addition map + : R × R ! R and a multiplication map × : R × R ! R that satisfy the following conditions. 1. R is a group under addition. 3 CRing Project, Chapter 1 2. The multiplication map is commutative and distributes over addition. This means that x × (y + z) = x × y + x × z and x × y = y × x. 3. There is a unit (or identity element), denoted by 1, such that 1 × x = x for all x 2 R. We shall typically write xy for x × y. Given a ring, a subring is a subset that contains the identity element and is closed under addition and multiplication. A noncommutative (i.e. not necessarily commutative) ring is one satisfying the above con- ditions, except possibly for the commutativity requirement xy = yx. For instance, there is a noncommutative ring of 2-by-2 matrices over C. We shall not work too much with noncommuta- tive rings in the sequel, though many of the basic results (e.g. on modules) do generalize. Example 1.2 Z is the simplest example of a ring. Exercise 1.1 Let R be a commutative ring. Show that the set of polynomials in one variable over R is a commutative ring R[x]. Give a rigorous definition of this. Example 1.3 For any ring R, we can consider the polynomial ring R[x1; : : : ; xn] which con- sists of the polynomials in n variables with coefficients in R. This can be defined inductively as (R[x1; : : : ; xn−1])[xn], where the procedure of adjoining a single variable comes from the previous ?? 1.1. We shall see a more general form of this procedure in Example 1.9. Exercise 1.2 If R is a commutative ring, recall that an invertible element (or, somewhat confusingly, a unit) u 2 R is an element such that there exists v 2 R with uv = 1. Prove that v is necessarily unique. Exercise 1.3 Let X be a set and R a ring. The set RX of functions f : X ! R is a ring. 1.2 The category of rings The class of rings forms a category. Its morphisms are called ring homomorphisms. Definition 1.4 A ring homomorphism between two rings R and S as a map f : R ! S that respects addition and multiplication. That is, 1. f(1R) = 1S, where 1R and 1S are the respective identity elements. 2. f(a + b) = f(a) + f(b) for a; b 2 R. 3. f(ab) = f(a)f(b) for a; b 2 R. There is thus a category Ring whose objects are commutative rings and whose morphisms are ring-homomorphisms. The philosophy of Grothendieck, as expounded in his EGA [GD], is that one should always do things in a relative context. This means that instead of working with objects, one should work with morphisms of objects. Motivated by this, we introduce: Definition 1.5 Given a ring A, an A-algebra is a ring R together with a morphism of rings (a structure morphism) A ! R. There is a category of A-algebras, where a morphism between A-algebras is a ring-homomorphism that is required to commute with the structure morphisms. 4 CRing Project, Chapter 1 So if R is an A-algebra, then R is not only a ring, but there is a way to multiply elements of R by elements of A (namely, to multiply a 2 A with r 2 R, take the image of a in R, and multiply that by r). For instance, any ring is an algebra over any subring. We can think of an A-algebra as an arrow A ! R, and a morphism from A ! R to A ! S as a commutative diagram R / S _@@ ? @@ @@ @ A This is a special case of the undercategory construction. If B is an A-algebra and C a B-algebra, then C is an A-algebra in a natural way. Namely, by assumption we are given morphisms of rings A ! B and B ! C, so composing them gives the structure morphism A ! C of C as an A-algebra. Example 1.6 Every ring is a Z-algebra in a natural and unique way. There is a unique map (of rings) Z ! R for any ring R because a ring-homomorphism is required to preserve the identity. In fact, Z is the initial object in the category of rings: this is a restatement of the preceding discussion. Example 1.7 If R is a ring, the polynomial ring R[x] is an R-algebra in a natural manner. Each element of R is naturally viewed as a \constant polynomial." Example 1.8 C is an R-algebra. Here is an example that generalizes the case of the polynomial ring. Example 1.9 If R is a ring and G a commutative monoid,1 then the set R[G] of formal finite P sums rigi with ri 2 R; gi 2 G is a commutative ring, called the moniod ring or group ring P when G is a group. Alternatively, we can think of elements of R[G] as infinite sums g2G rgg with R-coefficients, such that almost all the rg are zero. We can define the multiplication law such that 0 1 X X X X rgg sgg = @ rgsg0 A h: h gg0=h This process is called convolution. We can think of the multiplication law as extended the group multiplication law (because the product of the ring-elements corresponding to g; g0 is the ring element corresponding to gg0 2 G). The case of G = Z≥0 is the polynomial ring. In some cases, we can extend this notion to formal infinite sums, as in the case of the formal power series ring; see ?? below. Exercise 1.4 The ring Z is an initial object in the category of rings. That is, for any ring R, there is a unique morphism of rings Z ! R. We discussed this briefly earlier; show more generally that A is the initial object in the category of A-algebras for any ring A. Exercise 1.5 The ring where 0 = 1 (the zero ring) is a final object in the category of rings. That is, every ring admits a unique map to the zero ring. Exercise 1.6 Let C be a category and F : C! Sets a covariant functor. Recall that F is said to be corepresentable if F is naturally isomorphic to X ! HomC(U; X) for some object U 2 C.