🧮Commutative Algebra Unit 2 – Ring Homomorphisms and Quotient Rings

Key Concepts and Definitions

• A ring is a set $R$ equipped with two binary operations, addition and multiplication, satisfying certain axioms such as associativity, distributivity, and the existence of additive and multiplicative identities
• An ideal $I$ of a ring $R$ is a subring of $R$ that "absorbs" elements under multiplication: for all $r \in R$ and $a \in I$, $ra \in I$
  • Examples of ideals include the zero ideal $\{0\}$, the entire ring $R$, and the set of even integers in $\mathbb{Z}$
• A ring homomorphism is a function $f: R \to S$ between two rings that preserves the ring structure: $f(a+b) = f(a) + f(b)$ and $f(ab) = f(a)f(b)$ for all $a,b \in R$
• The kernel of a ring homomorphism $f: R \to S$ is the set of elements in $R$ that map to the additive identity in $S$: $\ker(f) = \{r \in R : f(r) = 0_S\}$
• The image of a ring homomorphism $f: R \to S$ is the set of elements in $S$ that are "hit" by $f$: $\operatorname{im}(f) = \{f(r) : r \in R\}$
• A quotient ring, denoted $R/I$, is a ring constructed from a ring $R$ and an ideal $I$ by "collapsing" elements that differ by an element of $I$
  • The elements of $R/I$ are cosets of the form $a + I = \{a + i : i \in I\}$

Ring Homomorphisms Explained

• A ring homomorphism is a structure-preserving map between two rings that respects both the additive and multiplicative operations
• Ring homomorphisms generalize the notion of group homomorphisms, which preserve the group operation, to the context of rings with two operations
• For a function $f: R \to S$ to be a ring homomorphism, it must satisfy the following properties for all $a,b \in R$:
  • $f(a+b) = f(a) + f(b)$ (preserves addition)
  • $f(ab) = f(a)f(b)$ (preserves multiplication)
  • $f(1_R) = 1_S$ (maps the multiplicative identity to the multiplicative identity)
• Examples of ring homomorphisms include the inclusion map from $\mathbb{Z}$ to $\mathbb{Q}$, the projection map from $\mathbb{R}[x]$ to $\mathbb{R}$ sending a polynomial to its constant term, and the complex conjugation map from $\mathbb{C}$ to $\mathbb{C}$
• Ring homomorphisms allow us to study the relationships between different rings and transfer properties from one ring to another

Properties of Ring Homomorphisms

• Ring homomorphisms preserve many algebraic properties, enabling us to study rings by examining their homomorphic images
• If $f: R \to S$ is a ring homomorphism, then:
  • $f(0_R) = 0_S$ (the zero element is mapped to the zero element)
  • $f(-a) = -f(a)$ for all $a \in R$ (negatives are preserved)
  • If $a$ is a unit in $R$, then $f(a)$ is a unit in $S$ (units are mapped to units)
  • If $R$ is commutative, then $\operatorname{im}(f)$ is a commutative subring of $S$
• Kernels of ring homomorphisms are always ideals of the domain ring
  • Conversely, every ideal of a ring $R$ is the kernel of some ring homomorphism (the natural projection $R \to R/I$)
• The composition of two ring homomorphisms is again a ring homomorphism
• Isomorphisms between rings are bijective ring homomorphisms; they preserve all ring-theoretic properties

Kernel and Image of Ring Homomorphisms

• The kernel of a ring homomorphism $f: R \to S$ is the preimage of the zero element in $S$: $\ker(f) = \{r \in R : f(r) = 0_S\}$
  • $\ker(f)$ is always an ideal of $R$, not just a subring
• The image of a ring homomorphism $f: R \to S$ is the set of elements in $S$ that are "reachable" by $f$: $\operatorname{im}(f) = \{f(r) : r \in R\}$
  • $\operatorname{im}(f)$ is always a subring of $S$, and if $R$ is commutative, then $\operatorname{im}(f)$ is a commutative subring
• The kernel and image of a ring homomorphism are related by the First Isomorphism Theorem: $R/\ker(f) \cong \operatorname{im}(f)$
  • This theorem allows us to understand the structure of the image of a ring homomorphism by studying the quotient of the domain by the kernel
• Examples:
  • For the inclusion map $i: \mathbb{Z} \to \mathbb{Q}$, $\ker(i) = \{0\}$ and $\operatorname{im}(i) = \mathbb{Z}$
  • For the projection map $\pi: \mathbb{R}[x] \to \mathbb{R}$ sending a polynomial to its constant term, $\ker(\pi) = (x)$ (the ideal generated by $x$) and $\operatorname{im}(\pi) = \mathbb{R}$

Introduction to Quotient Rings

• Given a ring $R$ and an ideal $I$, the quotient ring $R/I$ is a new ring constructed by "collapsing" elements of $R$ that differ by an element of $I$
• The elements of $R/I$ are cosets of the form $a + I = \{a + i : i \in I\}$, where $a \in R$
  • Two elements $a,b \in R$ are in the same coset if and only if $a-b \in I$
• Addition and multiplication in $R/I$ are defined by operating on representatives of the cosets:
  • $(a+I) + (b+I) = (a+b) + I$
  • $(a+I)(b+I) = ab + I$
• The zero element of $R/I$ is the coset $I$ itself, and the multiplicative identity is the coset $1 + I$
• The natural projection map $\pi: R \to R/I$ sending $a$ to $a+I$ is a surjective ring homomorphism with kernel $I$
• Examples:
  • $\mathbb{Z}/n\mathbb{Z}$ is the ring of integers modulo $n$, obtained by quotienting $\mathbb{Z}$ by the ideal $(n) = \{nk : k \in \mathbb{Z}\}$
  • $\mathbb{R}[x]/(x^2+1)$ is the ring of complex numbers $\mathbb{C}$, obtained by quotienting the polynomial ring $\mathbb{R}[x]$ by the ideal generated by $x^2+1$

Constructing Quotient Rings

• To construct the quotient ring $R/I$, we start with a ring $R$ and an ideal $I$ and define an equivalence relation on $R$ by $a \sim b$ if and only if $a-b \in I$
  • This equivalence relation partitions $R$ into cosets of the form $a+I$, which become the elements of $R/I$
• The coset operations in $R/I$ are well-defined because they do not depend on the choice of representatives:
  • If $a+I = a'+I$ and $b+I = b'+I$, then $(a+I) + (b+I) = (a'+I) + (b'+I)$ and $(a+I)(b+I) = (a'+I)(b'+I)$
• The quotient ring $R/I$ inherits many properties from the original ring $R$:
  • If $R$ is commutative, then $R/I$ is commutative
  • If $R$ is a domain and $I$ is a prime ideal, then $R/I$ is a domain
  • If $R$ is a principal ideal domain and $I$ is a maximal ideal, then $R/I$ is a field
• The natural projection $\pi: R \to R/I$ is a surjective ring homomorphism with kernel $I$, so by the First Isomorphism Theorem, $R/\ker(\pi) \cong \operatorname{im}(\pi) = R/I$

Isomorphism Theorems for Rings

• The isomorphism theorems for rings generalize the isomorphism theorems for groups and provide a powerful tool for understanding the structure of rings and their homomorphisms
• The First Isomorphism Theorem states that if $f: R \to S$ is a ring homomorphism, then $R/\ker(f) \cong \operatorname{im}(f)$
  • This theorem relates the kernel and image of a ring homomorphism and allows us to study the image by examining the quotient of the domain by the kernel
• The Second Isomorphism Theorem states that if $I$ is an ideal of a ring $R$ and $J$ is a subring of $R$ containing $I$, then $J/I$ is an ideal of $R/I$ and $(R/I)/(J/I) \cong R/J$
  • This theorem relates the quotients of nested subrings and ideals
• The Third Isomorphism Theorem (also known as the Correspondence Theorem) states that if $I$ and $J$ are ideals of a ring $R$ with $I \subseteq J$, then there is a one-to-one correspondence between the ideals of $R/I$ containing $J/I$ and the ideals of $R$ containing $J$, given by $K \mapsto \pi^{-1}(K)$, where $\pi: R \to R/I$ is the natural projection
  • This theorem establishes a bijection between certain ideals of a ring and ideals of its quotient ring

Applications and Examples

• Quotient rings have numerous applications in algebra and beyond, providing a way to construct new rings with desired properties and to solve equations by working in simpler quotient structures
• Polynomial rings and their quotients are central to algebraic geometry, as they provide a way to study geometric objects using algebraic techniques
  • For example, the quotient ring $\mathbb{C}[x,y]/(y^2-x^3-x)$ corresponds to the elliptic curve $y^2 = x^3+x$ in the plane
• Quotient rings are used in number theory to study congruences and solve Diophantine equations
  • For instance, the Chinese Remainder Theorem can be interpreted as an isomorphism between a quotient ring and a product of quotient rings: $\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/p_1^{k_1}\mathbb{Z} \times \cdots \times \mathbb{Z}/p_r^{k_r}\mathbb{Z}$ when $n = p_1^{k_1} \cdots p_r^{k_r}$ is a prime factorization
• In cryptography, quotient rings of polynomial rings over finite fields are used in the construction of error-correcting codes and public-key cryptosystems
  • The quotient ring $\mathbb{F}_2[x]/(x^n+1)$ is used in the Rijndael (AES) encryption algorithm for secure communication
• Quotient rings also appear in the study of operator algebras and representation theory, where they provide a way to construct representations of groups and algebras
  • For example, the group algebra $\mathbb{C}[G]$ of a finite group $G$ can be studied using its quotients by ideals corresponding to irreducible representations of $G$