Ring Theory 01: Rings, Ideals, Quotients, and Modules

Ring theory / 01

Rings, ideals, quotients, and modules

The affine-scheme dictionary starts with ordinary algebra: rings behave like functions, ideals behave like systems of equations, quotient rings impose those equations, and modules behave like algebraic families of linear data.

Atiyah and Macdonald begin with rings, ideals, and modules because all later geometry is expressed through them. This article compresses the elementary set language into the places where it matters and then builds the algebraic vocabulary used throughout the series.

Sets, functions, and inverse images

A function \(f:X\to Y\) sends each element of \(X\) to one element of \(Y\). If \(B\subseteq Y\), the inverse image is

\[f^{-1}(B)=\{x\in X:f(x)\in B\}.\]

Inverse images are structurally better behaved than direct images: they preserve arbitrary unions, arbitrary intersections, and complements. This simple fact explains why continuous maps are defined by inverse images of open sets, and it foreshadows why a ring homomorphism \(A\to B\) induces a map

\[\operatorname{Spec}B\to\operatorname{Spec}A\]

by inverse image of prime ideals.

Commutative rings

A commutative ring \(A\) is an abelian group under addition, has an associative and commutative multiplication, has an identity element \(1\), and satisfies distributivity:

\[a(b+c)=ab+ac.\]

The standing examples are

\[\mathbb Z,\qquad k,\qquad k[x],\qquad k[x_1,\ldots,x_n],\qquad k[x,y]/(xy).\]

The geometric interpretation is that \(A\) is a ring of functions on a hypothetical space. Addition and multiplication are pointwise operations, but the space must be reconstructed from algebra.

Example: functions first, space later

The ring \(\mathbb C[x]\) is naturally the polynomial functions on the complex affine line. A point \(a\in\mathbb C\) gives an evaluation homomorphism

\[\operatorname{ev}_a:\mathbb C[x]\to\mathbb C,\qquad f\mapsto f(a).\]

Its kernel is the maximal ideal \((x-a)\). Scheme theory reverses this observation: start from a ring \(A\), then define the points by suitable ideals.

Ring homomorphisms

A ring homomorphism \(\varphi:A\to B\) satisfies

\[\varphi(a+b)=\varphi(a)+\varphi(b),\qquad \varphi(ab)=\varphi(a)\varphi(b),\qquad \varphi(1_A)=1_B.\]

The kernel

\[\ker\varphi=\{a\in A:\varphi(a)=0\}\]

is an ideal, and the image is a subring of \(B\). In geometry, a homomorphism \(\varphi:A\to B\) is read contravariantly as a map from the space of \(B\) to the space of \(A\). Functions pull back along maps; therefore rings of functions reverse arrows.

Ideals

An ideal \(I\subseteq A\) is an additive subgroup closed under multiplication by arbitrary elements of \(A\):

\[a,b\in I\Rightarrow a-b\in I,\qquad r\in A,\ a\in I\Rightarrow ra\in I.\]

The ideal generated by \(f_1,\ldots,f_n\) is

\[(f_1,\ldots,f_n) = \left\{\sum_{i=1}^n a_i f_i:a_i\in A\right\}.\]

Geometrically, an ideal is a list of functions required to vanish. The larger the ideal, the smaller the expected zero set.

Order reversal

If \(I\subseteq J\), then every prime containing \(J\) contains \(I\), so

\[V(J)\subseteq V(I).\]

More equations mean fewer points. This order reversal is everywhere in algebraic geometry.

Quotient rings

For an ideal \(I\subseteq A\), the quotient ring \(A/I\) identifies two elements when their difference lies in \(I\). Its elements are residue classes \(a+I\), and multiplication is

\[(a+I)(b+I)=ab+I.\]

The quotient map

\[\pi:A\to A/I\]

is universal among ring homomorphisms out of \(A\) that kill \(I\): if \(\varphi:A\to B\) satisfies \(I\subseteq\ker\varphi\), then \(\varphi\) factors uniquely through \(A/I\).

Example: the crossing axes

Let

\[A=k[x,y]/(xy).\]

The equation \(xy=0\) says that every point lies on the \(x\)-axis or the \(y\)-axis. The quotient has zero divisors: the classes of \(x\) and \(y\) are nonzero but their product is zero. This algebraic zero-divisor records reducible geometry.

Units, zero divisors, and nilpotents

An element \(u\in A\) is a unit if there exists \(v\in A\) with \(uv=1\). A nonzero element \(a\) is a zero divisor if \(ab=0\) for some nonzero \(b\). An element \(n\) is nilpotent if \(n^N=0\) for some \(N>0\).

Nilpotents are invisible to ordinary point evaluation over fields, but they are not meaningless. The ring

\[k[\epsilon]/(\epsilon^2)\]

has one classical point but remembers a first-order infinitesimal direction. Scheme theory keeps this information.

The nilradical is

\[\sqrt{(0)}=\{a\in A:a^N=0\text{ for some }N\},\]

and more generally the radical of an ideal is

\[\sqrt I=\{a\in A:a^N\in I\text{ for some }N>0\}.\]

Atiyah and Macdonald emphasize radicals because \(V(I)=V(\sqrt I)\) as sets. The sheaf and scheme structure, however, can distinguish \(I\) from \(\sqrt I\).

Modules

An \(A\)-module \(M\) is an abelian group with scalar multiplication \(A\times M\to M\) satisfying the usual linearity axioms. Modules over a ring are the correct replacement for vector spaces when scalars need not form a field.

Examples:

1. \(A\) is an \(A\)-module over itself.
2. Every ideal \(I\subseteq A\) is an \(A\)-module.
3. A quotient \(A/I\) is an \(A\)-module.
4. If \(A=\mathbb C[x]\) and \(a\in\mathbb C\), then \(A/(x-a)\) is a module supported at one closed point.
5. If \(A=\mathbb Z\) and \(p\) is a rational prime, then \(\mathbb Z/(p)\) is a module supported at the arithmetic closed point \((p)\).

These examples are the affine algebraic source of skyscraper-like sheaves. After applying the tilde construction, \(\mathbb C[x]/(x-a)\) has nonzero stalk only over primes containing \((x-a)\), hence only at the closed point \((x-a)\). Similarly, \(\mathbb Z/(p)\) has nonzero stalk only at \((p)\).

Exact sequences

A sequence of \(A\)-modules

\[M'\xrightarrow{u}M\xrightarrow{v}M''\]

is exact at \(M\) if \(\operatorname{im}u=\ker v\). A short exact sequence

\[0\to M'\to M\to M''\to 0\]

says that \(M’\) is a submodule of \(M\) and \(M’’\) is the corresponding quotient.

Exactness is the bookkeeping language for relations. Later, localization and the tilde construction preserve exact sequences, which is one reason quasi-coherent sheaves are tractable on affine schemes.

Tensor products and change of scalars

If \(A\to B\) is a ring homomorphism and \(M\) is an \(A\)-module, then

\[M\otimes_A B\]

is the \(B\)-module obtained by extending scalars from \(A\) to \(B\). Localization is the most important case:

\[M_f=M\otimes_A A_f.\]

This formula will become the definition of sections of \(\widetilde M\) over the basic open set \(D(f)\).

Common confusion

An \(A\)-module is not automatically a vector bundle. Free modules correspond to trivial vector bundles, finitely generated projective modules correspond to algebraic vector bundles on affine schemes, and arbitrary finite modules correspond to coherent sheaves that may have torsion or singular support.