Quotient ring
Introduction
In abstract algebra, a quotient ring (also known as a factor ring) is a construction that generalizes the notion of forming a quotient group. It is a way of creating a new ring from an existing ring and an ideal within that ring. The quotient ring is a fundamental concept in ring theory, a branch of mathematics that studies algebraic structures known as rings.
Definition
Let \( R \) be a ring and \( I \) be an ideal of \( R \). The quotient ring, denoted by \( R/I \), is the set of cosets of \( I \) in \( R \). Each coset is of the form \( r + I \) where \( r \in R \). The operations of addition and multiplication in \( R/I \) are defined as follows:
- **Addition**: \((r + I) + (s + I) = (r + s) + I\)
- **Multiplication**: \((r + I) \cdot (s + I) = (r \cdot s) + I\)
The quotient ring \( R/I \) inherits the ring structure from \( R \), making it a well-defined ring.
Properties
Ring Homomorphism
There is a natural ring homomorphism from \( R \) to \( R/I \) given by the map \( \phi: R \to R/I \) where \( \phi(r) = r + I \). This map is surjective, and its kernel is exactly the ideal \( I \). This homomorphism is crucial in understanding the structure of quotient rings.
First Isomorphism Theorem
The First Isomorphism Theorem for rings states that if \( \phi: R \to S \) is a ring homomorphism with kernel \( K \), then \( R/K \cong \phi(R) \). This theorem implies that every quotient ring \( R/I \) is isomorphic to some subring of \( R \).
Ideal Correspondence
There is a one-to-one correspondence between the ideals of \( R \) that contain \( I \) and the ideals of \( R/I \). If \( J \) is an ideal of \( R \) containing \( I \), then the corresponding ideal in \( R/I \) is \( J/I \).
Prime and Maximal Ideals
A prime ideal \( P \) in \( R \) leads to a prime ideal \( P/I \) in \( R/I \). Similarly, a maximal ideal \( M \) in \( R \) leads to a maximal ideal \( M/I \) in \( R/I \). This correspondence is significant in the study of the structure of rings and their quotient rings.
Examples
Polynomial Rings
Consider the ring of polynomials \( \mathbb{R}[x] \) and the ideal \( (x^2 + 1) \). The quotient ring \( \mathbb{R}[x]/(x^2 + 1) \) is isomorphic to the complex numbers \( \mathbb{C} \). This is because the polynomial \( x^2 + 1 \) has no real roots, and the quotient ring effectively introduces a root \( i \) such that \( i^2 = -1 \).
Integer Modulo n
The ring \( \mathbb{Z}/n\mathbb{Z} \) is a quotient ring where \( \mathbb{Z} \) is the ring of integers and \( n\mathbb{Z} \) is the ideal consisting of all multiples of \( n \). This quotient ring is commonly known as the ring of integers modulo \( n \).
Applications
Algebraic Geometry
In algebraic geometry, quotient rings are used to study algebraic varieties. The coordinate ring of an algebraic variety can be viewed as a quotient ring of a polynomial ring.
Commutative Algebra
Quotient rings play a crucial role in commutative algebra, particularly in the study of local rings and localization. They help in understanding the local properties of rings and modules.
Module Theory
In module theory, quotient rings are used to construct quotient modules. If \( M \) is an \( R \)-module and \( N \) is a submodule, then the quotient module \( M/N \) can be viewed as an \( R/I \)-module where \( I \) is an ideal of \( R \).