Math Problem Statement

What is an ideal in ring theory and example

Solution

In ring theory, an ideal is a special subset of a ring that allows the construction of quotient rings, analogous to how normal subgroups allow for quotient groups in group theory.

Definition:

Given a ring RR, a subset II of RR is called an ideal if:

  1. Additive closure: II is closed under addition, i.e., for all a,bIa, b \in I, a+bIa + b \in I.
  2. Multiplicative closure: For every aIa \in I and rRr \in R, both rar \cdot a and ara \cdot r (in the case of non-commutative rings) are in II.

This means that multiplying an element of the ring with an element from the ideal always results in an element of the ideal.

Types of Ideals:

  1. Left Ideal: Closed under multiplication on the left, i.e., raIr \cdot a \in I for all rRr \in R and aIa \in I.
  2. Right Ideal: Closed under multiplication on the right, i.e., arIa \cdot r \in I for all rRr \in R and aIa \in I.
  3. Two-sided Ideal: Closed under multiplication on both sides, i.e., raIr \cdot a \in I and arIa \cdot r \in I for all rRr \in R and aIa \in I.

In commutative rings, left ideals, right ideals, and two-sided ideals coincide and are simply called ideals.

Example:

Consider the ring Z\mathbb{Z}, the set of integers with usual addition and multiplication. The set of even integers 2Z={,4,2,0,2,4,}2\mathbb{Z} = \{ \dots, -4, -2, 0, 2, 4, \dots \} is an ideal in Z\mathbb{Z}.

  1. It is closed under addition (the sum of any two even numbers is still even).
  2. For any integer nn and any even number 2m2m, the product n(2m)=2(nm)n \cdot (2m) = 2(nm) is still an even number, so the ideal is closed under multiplication by elements from the ring Z\mathbb{Z}.

This makes 2Z2\mathbb{Z} a two-sided ideal in Z\mathbb{Z}.

Key Insight:

Ideals are used to define quotient rings, much like normal subgroups allow the construction of quotient groups. If II is an ideal of a ring RR, we can form the quotient ring R/IR/I, which generalizes the idea of modular arithmetic.

Would you like further clarification or examples?


Here are 5 relative questions to expand on this topic:

  1. How are ideals related to kernels of ring homomorphisms?
  2. What is the difference between a prime ideal and a maximal ideal?
  3. How does the construction of a quotient ring work in practice?
  4. Can you explain what an ideal in a non-commutative ring looks like?
  5. What role do principal ideals play in the structure of rings?

Tip: Always check if an ideal is two-sided, especially in non-commutative rings, as this affects whether you can form a well-defined quotient ring.

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Math Problem Analysis

Mathematical Concepts

Ring Theory
Ideals
Quotient Rings

Formulas

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Theorems

Ideal definition in Ring Theory
Quotient Ring construction

Suitable Grade Level

University Level (Undergraduate Algebra)