8.3 Ideals and Quotient Rings

Ideals and quotient rings are key concepts in abstract algebra. They help us understand ring structures and create new rings from existing ones. Ideals are special subrings that absorb multiplication, while quotient rings are formed by "dividing" a ring by an ideal.

These concepts are crucial for studying ring homomorphisms and their kernels. They also allow us to classify rings based on their ideal structures, leading to important distinctions like prime and maximal ideals. Understanding these ideas is essential for deeper explorations in abstract algebra.

Ideals and Quotient Rings

Defining Ideals and Quotient Rings

• An ideal of a ring $R$ is a subring $I$ of $R$ such that for every $r$ in $R$ and every $a$ in $I$, both $ra$ and $ar$ are in $I$
• A quotient ring, also known as a factor ring, is a construction to produce a new ring from a given ring and a two-sided ideal in that ring
• For a ring $R$ and a two-sided ideal $I$ in $R$, the quotient ring is denoted by $R/I$ and is the set of cosets of $I$ in $R$, where addition and multiplication are defined on the cosets
• The elements of the quotient ring $R/I$ are the cosets $a + I = \{a + r \mid r \text{ is in } I\}$, where $a$ is in $R$
• In the quotient ring $R/I$, addition is defined by $(a + I) + (b + I) = (a + b) + I$, and multiplication is defined by $(a + I)(b + I) = ab + I$, for any $a$ and $b$ in $R$

Equivalence Relations and Coset Operations

• To construct a quotient ring $R/I$, define an equivalence relation on $R$ by $a \sim b$ if and only if $a - b$ is in $I$, for any $a$ and $b$ in $R$
• The equivalence classes under this relation are the cosets of $I$ in $R$, which form the elements of the quotient ring $R/I$
• Define addition and multiplication operations on the cosets as $(a + I) + (b + I) = (a + b) + I$ and $(a + I)(b + I) = ab + I$, respectively
• Verify that the quotient ring $R/I$ is indeed a ring under these operations, inheriting properties from the original ring $R$ (associativity, distributivity, identity elements, and inverse elements)

Types of Ideals

Prime and Maximal Ideals

• A prime ideal $P$ of a commutative ring $R$ is a proper ideal such that for any two elements $a$, $b$ in $R$, if their product $ab$ is in $P$, then $a$ is in $P$ or $b$ is in $P$
• A maximal ideal $M$ of a ring $R$ is a proper ideal that is not strictly contained in any other proper ideal of $R$
• Every maximal ideal in a commutative ring is a prime ideal, but the converse is not always true (there can be prime ideals that are not maximal)
• The zero ideal $\{0\}$ and the entire ring $R$ are always ideals of $R$, but they are not considered prime or maximal unless $R$ is the zero ring $\{0\}$

Examples of Prime and Maximal Ideals

• In the ring of integers $\mathbb{Z}$, the ideal $p\mathbb{Z}$ generated by a prime number $p$ is both a prime and maximal ideal
• In the polynomial ring $\mathbb{R}[x]$, the ideal $(x^2 + 1)$ is a maximal ideal, as the quotient ring $\mathbb{R}[x]/(x^2 + 1)$ is isomorphic to the field of complex numbers $\mathbb{C}$

Constructing Quotient Rings

Steps to Construct a Quotient Ring

• To construct a quotient ring $R/I$, first identify the ring $R$ and the ideal $I$ within $R$
• Define an equivalence relation on $R$ by $a \sim b$ if and only if $a - b$ is in $I$, for any $a$ and $b$ in $R$
• The equivalence classes under this relation are the cosets of $I$ in $R$, which form the elements of the quotient ring $R/I$
• Define addition and multiplication operations on the cosets as $(a + I) + (b + I) = (a + b) + I$ and $(a + I)(b + I) = ab + I$, respectively
• Verify that the quotient ring $R/I$ is indeed a ring under these operations, inheriting properties from the original ring $R$

Examples of Quotient Ring Constructions

• The quotient ring $\mathbb{Z}/n\mathbb{Z}$ (also denoted as $\mathbb{Z}_n$) is constructed from the ring of integers $\mathbb{Z}$ and the ideal $n\mathbb{Z}$ generated by a positive integer $n$
• The quotient ring $\mathbb{R}[x]/(x^2 + 1)$ is constructed from the polynomial ring $\mathbb{R}[x]$ and the ideal $(x^2 + 1)$ generated by the polynomial $x^2 + 1$

Properties of Ideals and Quotient Rings

Kernels and Homomorphisms

• Prove that the kernel of a ring homomorphism is an ideal of the domain ring
• Use the fundamental theorem of homomorphisms to prove that if $I$ is an ideal of a ring $R$, then the quotient ring $R/I$ is isomorphic to the image of the natural projection map $\pi: R \to R/I$ defined by $\pi(a) = a + I$ for all $a$ in $R$

Quotient Ring Structures

• Show that the quotient ring $R/I$ is a field if and only if $I$ is a maximal ideal of $R$
• Demonstrate that the quotient ring $R/I$ is an integral domain if and only if $I$ is a prime ideal of $R$
• Prove that the characteristic of the quotient ring $R/I$ is the smallest positive integer $n$ such that $n \cdot 1$ is in $I$, where $1$ is the multiplicative identity of $R$
• Establish that the quotient ring $R/I$ is a zero ring (i.e., it has only one element) if and only if $I = R$

Examples of Ideal and Quotient Ring Properties

• The quotient ring $\mathbb{Z}/p\mathbb{Z}$ (also denoted as $\mathbb{Z}_p$) is a field for any prime number $p$, as $p\mathbb{Z}$ is a maximal ideal in $\mathbb{Z}$
• The quotient ring $\mathbb{Z}/6\mathbb{Z}$ (also denoted as $\mathbb{Z}_6$) is not an integral domain, as $6\mathbb{Z}$ is not a prime ideal in $\mathbb{Z}$ (since $2 \cdot 3 = 6$, but neither $2$ nor $3$ is in $6\mathbb{Z}$)

Fundamental Theorem of Homomorphisms

Statement and Applications

• The fundamental theorem of homomorphisms states that if $f: R \to S$ is a surjective ring homomorphism with kernel $I$, then the quotient ring $R/I$ is isomorphic to $S$
• Apply the fundamental theorem to show that if $f: R \to S$ is a surjective ring homomorphism with kernel $I$, then there exists a unique isomorphism $g: R/I \to S$ such that $f = g \circ \pi$, where $\pi: R \to R/I$ is the natural projection map
• Utilize the fundamental theorem to solve problems involving the structure and properties of quotient rings, such as determining the characteristics or classifying types of quotient rings based on the properties of the original ring and the ideal

Examples of Applying the Fundamental Theorem

• Use the fundamental theorem to prove that the quotient ring $\mathbb{R}[x]/(x^2 + 1)$ is isomorphic to the field of complex numbers $\mathbb{C}$, by considering the evaluation homomorphism $\mathbb{R}[x] \to \mathbb{C}$ defined by $f(p(x)) = p(i)$
• Apply the fundamental theorem to show that the quotient ring $\mathbb{Z}[i]/(3)$ is isomorphic to the finite field $\mathbb{F}_9$, by considering the natural projection homomorphism $\mathbb{Z}[i] \to \mathbb{Z}[i]/(3)$ and the isomorphism between $\mathbb{Z}[i]/(3)$ and $\mathbb{F}_9$