1.3 Polynomial rings and ideals

Polynomial rings and ideals are the building blocks of algebraic geometry. They provide a powerful framework for studying geometric objects through algebraic equations, bridging the gap between algebra and geometry.

In this context, we'll explore how polynomial rings encode geometric information and how ideals represent sets of solutions. We'll see how these concepts form the foundation for understanding algebraic varieties and their properties.

Polynomial rings and their properties

Definition and notation

• A polynomial ring, denoted as $R[x]$, is a ring formed by polynomials with coefficients from a ring $R$
• The elements of a polynomial ring are polynomials, and the operations of addition and multiplication are performed on these polynomials
  • For example, if $R$ is the ring of integers, then $R[x]$ consists of polynomials with integer coefficients, such as $3x^2 + 2x - 1$

Properties inherited from the coefficient ring

• Polynomial rings inherit many properties from the coefficient ring $R$, such as commutativity and the existence of identity elements for addition and multiplication
  • If $R$ is commutative, then $R[x]$ is also commutative, meaning that for any polynomials $f(x)$ and $g(x)$ in $R[x]$, $f(x)g(x) = g(x)f(x)$
  • The zero polynomial and the constant polynomial 1 serve as the additive and multiplicative identity elements in $R[x]$, respectively
• If $R$ is a field, then $R[x]$ is a principal ideal domain (PID), meaning that every ideal in $R[x]$ is generated by a single polynomial
  • For instance, if $R$ is the field of real numbers, then every ideal in $R[x]$ is of the form $(f(x))$, where $f(x)$ is a polynomial in $R[x]$

Degree of a polynomial

• The degree of a polynomial is the highest power of the variable in the polynomial, and it plays a crucial role in determining the properties of the polynomial ring
  • For example, the polynomial $3x^2 + 2x - 1$ has degree 2, as the highest power of $x$ is 2
• Polynomials of the same degree can be compared and ordered based on their leading coefficients, allowing for the division algorithm and the notion of greatest common divisors (GCDs) in $R[x]$

Ideals in algebraic geometry

Definition and properties

• An ideal $I$ in a ring $R$ is a subset of $R$ that is closed under addition and multiplication by elements of $R$
  • For any $a, b \in I$ and $r \in R$, $a + b \in I$ and $ra \in I$
• Ideals generalize the concept of multiples in the ring of integers and allow for the study of congruences and quotient rings
  • In the ring of integers $\mathbb{Z}$, the ideal $(n)$ consists of all multiples of $n$, and the quotient ring $\mathbb{Z}/(n)$ represents the congruence classes modulo $n$

Correspondence with algebraic sets

• In algebraic geometry, ideals in polynomial rings are used to define algebraic sets, which are the solution sets of systems of polynomial equations
  • For example, the ideal $I = (x^2 + y^2 - 1, y - x^2)$ in $\mathbb{R}[x, y]$ defines the algebraic set $V(I)$, which is the unit circle intersected with the parabola $y = x^2$
• The correspondence between ideals and algebraic sets is a fundamental concept in algebraic geometry, known as the Zariski topology
  • Closed sets in the Zariski topology are precisely the algebraic sets, and the topology is defined by taking finite unions and arbitrary intersections of algebraic sets

Prime and maximal ideals

• Prime ideals, which are ideals $P$ such that for any $a, b \in R$, if $ab \in P$, then either $a \in P$ or $b \in P$, play a crucial role in understanding the geometry of algebraic sets
  • Prime ideals correspond to irreducible algebraic sets, which cannot be written as the union of two proper subsets
• Maximal ideals, which are ideals that are maximal with respect to inclusion, correspond to points in the affine space defined by the polynomial ring
  • For example, in $\mathbb{C}[x, y]$, the maximal ideal $(x - 1, y - 2)$ corresponds to the point $(1, 2)$ in the complex affine plane

Operations on ideals

Sum, product, and intersection

• The sum of two ideals $I$ and $J$, denoted as $I + J$, is the smallest ideal containing both $I$ and $J$
  • $I + J = \{a + b : a \in I, b \in J\}$
• The product of two ideals $I$ and $J$, denoted as $IJ$, is the ideal generated by all products of elements from $I$ and $J$
  • $IJ = \{\sum_{i=1}^n a_ib_i : a_i \in I, b_i \in J, n \in \mathbb{N}\}$
• The intersection of two ideals $I$ and $J$, denoted as $I \cap J$, is the largest ideal contained in both $I$ and $J$
  • $I \cap J = \{a : a \in I \text{ and } a \in J\}$

Quotient and radical

• The quotient of two ideals $I$ and $J$, denoted as $(I : J)$, is the ideal consisting of all elements $r$ in $R$ such that $rJ$ is a subset of $I$
  • $(I : J) = \{r \in R : rJ \subseteq I\}$
• The radical of an ideal $I$, denoted as $\sqrt{I}$, is the ideal consisting of all elements $r$ in $R$ such that some power of $r$ belongs to $I$
  • $\sqrt{I} = \{r \in R : r^n \in I \text{ for some } n \in \mathbb{N}\}$
• Operations on ideals allow for the manipulation and simplification of systems of polynomial equations in algebraic geometry
  • For instance, the radical of an ideal corresponds to the vanishing set of the ideal, which is the set of all points where all polynomials in the ideal evaluate to zero

Geometric meaning of ideals

Algebraic sets and vanishing ideals

• The algebraic set defined by an ideal $I$, denoted as $V(I)$, is the set of all points in the affine space that satisfy all the polynomial equations in $I$
  • $V(I) = \{(a_1, \ldots, a_n) \in k^n : f(a_1, \ldots, a_n) = 0 \text{ for all } f \in I\}$, where $k$ is the underlying field
• The ideal of an algebraic set $S$, denoted as $I(S)$, is the set of all polynomials that vanish on every point of $S$
  • $I(S) = \{f \in k[x_1, \ldots, x_n] : f(a_1, \ldots, a_n) = 0 \text{ for all } (a_1, \ldots, a_n) \in S\}$

Zariski topology and irreducible sets

• The Zariski topology on the affine space is defined by taking algebraic sets as the closed sets, establishing a correspondence between ideals and closed sets
  • The closure of a set in the Zariski topology is the smallest algebraic set containing it, and it corresponds to the radical of the ideal of the set
• Irreducible algebraic sets, which cannot be written as the union of two proper subsets, correspond to prime ideals in the polynomial ring
  • An algebraic set is irreducible if and only if its ideal is a prime ideal

Dimension and local properties

• The dimension of an algebraic set is related to the height of its corresponding prime ideal, providing a way to study the geometric properties of the set
  • The dimension of an irreducible algebraic set is the transcendence degree of its function field over the base field
• The local ring at a point in an algebraic set is obtained by localizing the polynomial ring at the maximal ideal corresponding to that point, allowing for the study of local properties of the set
  • The localization of a ring $R$ at a prime ideal $P$, denoted as $R_P$, consists of elements of the form $\frac{a}{b}$, where $a \in R$ and $b \in R \setminus P$, and it captures the local behavior of the algebraic set near the point corresponding to $P$