Rings, Modules, Fields

1 Rings

A ring is a monoid object in the category of abelian groups via the monoidal structure given by the tensor product \(\otimes \), the left adjoint of \(\Hom \). In other words, it is an abelian group \((R,+)\) with a bilinear multiplication operation \(*\) that is associative, and has an identity called \(1\). The category of rings is complete and cocomplete, its terminal object is the unique ring with \(1=0\), and is called the zero or trivial ring, and the initial object is the integers, \(\mathbb {Z}\). A property of rings is that \(0*a\) (often written \(0a\) for concision) is \(0\) for any \(a\).

The reason rings are important is quite similar to the reason groups are important. Groups arise naturally as the automorphisms of objects in a category, and similarly rings arise as the endomorphisms of objects in a preadditive category (i.e. one where the \(\Hom \)s are abelian groups and composition is bilinear). In other words, a ring is a one object preadditive category.

Thus it makes sense for a ring to “act" on a preadditive category. We call a functor from \(R\) to a preadditive category an \(R\)-module. If the category is unspecified, one usually assumes that it is the category of abelian groups, \(Ab\). If \(M\) is an \(R\)-module, \(m \in M, r \in R\), we write \(rm\) to mean the (bilinear) action of \(R\) on \(M\).

We will focus on commutative rings, as they are easier to understand than non-commutative rings. In fact, sometimes one even studies rings without \(1\), but we will again not think about this. From now on, commutative rings will be referred to as just rings.

Now for a commutative ring \(R\), analogously to the case of abelian groups, monoids in its category of modules are called \(R\)-algebras. For example, \(\ZZ \)-algebras are just rings, since \(\ZZ \)-modules are just abelian groups.

We can consider \(R\) to be an \(R\)-module in the natural way, the submodules of \(R\) are called ideals. In particular, these are nonempty subsets of \(R\) that are closed under multiplication by any element of \(R\) and under addition. Ideals are analogous to normal subgroups in that given a ring homomorphism \(R \to S\), the kernel is an ideal, and conversely given an ideal \(I\), we can form the quotient ring \(R/I\) by defining multiplication on cosets. Analogously to the case for groups, we have isomorphism theorems for rings. Note that we can write \((a,b,\dots )\) to mean the smallest ideal containing those elements, which we call the ideal generated by \(a,b,\dots \). Now a module is called Noetherian if every submodule is finitely generated, and a ring is called Noetherian if every ideal (submodule) is finitely generated.

One special kind of ideal \(\pp \) is called a prime ideal, which means that \(1 \notin \pp \), and if \(ab \in \pp \), then \(a \in \pp \) or \(b \in \pp \). An integral domain (sometimes just domain) is a nontrivial ring in which if \(ab = 0\), \(a=0\) or \(b = 0\), so we see that a prime ideal is exactly one where the quotient is an integral domain. Alternatively, an integral domain is a ring where \(0\) is a prime ideal. Yet another definition is a ring where there are no nonzero zero divisors, i.e. elements \(a\) such that \(ab = 0\), but \(b \neq 0\) for some \(b\). Note the nice property of prime ideals that the preimage of a prime ideal under a ring homomorphism is prime.

There is also the notion of a maximal ideal, which is an ideal \(I\subset R\) which is maximal in the poset of proper ideals, i.e those ideals not containing \(1\). By Zorn’s Lemma, maximal ideal always exist in nonzero rings. Analogously to prime ideals and integral domains, a field is a ring with two ideals. Equivalently it is a nonzero ring, where nonzero elements are units. Fields are then exactly the quotient of rings by a maximal ideal. In particular since units in a nonzero ring are not zero divisors, maximal ideals are prime.

There are many operations of ideals \(I,J\). For example, we can take meets \(I+J\) and joins \(I\cap J\), and products \(IJ\), which is the ideal generated by products of elements of \(I\) and \(J\). There is also the ideal quotient \((I:J) = \{r|rJ\subset I\}\).

It is always the case that \(IJ \subset I\cap J\), but they are not always equal. The following theorem gives a way to note when they are equal, and also realizes the quotient by an intersection as a pushout. It uses the notion of two ideals being comaximal, which means that their sum contains \(1\).

Proposition 1.1 (CRT). Suppose that \(I,J\subset R\) are ideals. Then \(R/(I\cap J)\) is the pullback of \(R/J\) and \(R/I\) along \(R/(I+J)\). If \(I,J\) are comaximal, \(IJ = (I+J)\), and \(R/IJ = R/I \times R/J\).

Proof. Given elements \([a],[b]\) in \(R/I, R/J\) that agree on \(R/(I+J)\), \(a = b + x+y\), with \(x,y \in I,J\) respectively. Then \(a-x=b+y\) is congruent mod \(I\) to a, and mod \(J\) to \(b\). Moreover, it is the unique such class mod \(I\cap J\), since if \(a-x'=b+y'\) were such a class, then \(x-x'=y-y' \equiv 0\) mod \(I\) and \(J\), and hence mod \(I\cap J\). Now if \(I,J\) are comaximal and \(a \in I\cap J\), then write \(1 = i+j \in I+J\), so that \(a = ai+aj\) is in \(IJ\). □