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 , a subset of is called an ideal if:
- Additive closure: is closed under addition, i.e., for all , .
- Multiplicative closure: For every and , both and (in the case of non-commutative rings) are in .
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:
- Left Ideal: Closed under multiplication on the left, i.e., for all and .
- Right Ideal: Closed under multiplication on the right, i.e., for all and .
- Two-sided Ideal: Closed under multiplication on both sides, i.e., and for all and .
In commutative rings, left ideals, right ideals, and two-sided ideals coincide and are simply called ideals.
Example:
Consider the ring , the set of integers with usual addition and multiplication. The set of even integers is an ideal in .
- It is closed under addition (the sum of any two even numbers is still even).
- For any integer and any even number , the product is still an even number, so the ideal is closed under multiplication by elements from the ring .
This makes a two-sided ideal in .
Key Insight:
Ideals are used to define quotient rings, much like normal subgroups allow the construction of quotient groups. If is an ideal of a ring , we can form the quotient ring , which generalizes the idea of modular arithmetic.
Would you like further clarification or examples?
Here are 5 relative questions to expand on this topic:
- How are ideals related to kernels of ring homomorphisms?
- What is the difference between a prime ideal and a maximal ideal?
- How does the construction of a quotient ring work in practice?
- Can you explain what an ideal in a non-commutative ring looks like?
- 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)