Alphabeta Math

Ideals, Quotient Rings and the Isomorphism Theorems for Rings

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✓ 19 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Left, right and two-sided ideals

Lemma

Ideal criteria and intersections of ideals

Definition

The ideal generated by a subset and principal ideals

Theorem

In a commutative ring, (S) consists of finite sums ∑ rᵢ sᵢ, and (a)=Ra

Definition

The sum I+J and product IJ of two-sided ideals

Theorem

The sum and product of two-sided ideals are two-sided ideals

Theorem

The kernel of a ring homomorphism is a two-sided ideal

Definition

The quotient ring R/I with (r+I)(s+I)=rs+I

Theorem

Multiplication of additive cosets is well defined if and only if the additive subgroup is a two-sided ideal

Theorem

For a two-sided ideal I, the additive cosets form a ring R/I with identity 1+I

Proposition

The canonical projection R→ R/I is a surjective ring homomorphism with kernel I

Proposition

For every n∈ℕ, the congruence-class ring ℤ/n is the quotient ring ℤ/nℤ

Theorem

A ring homomorphism whose kernel contains a two-sided ideal factors uniquely through the quotient ring

Theorem

First isomorphism theorem for rings: R/ker f congimf

Lemma

If S is a subring and I is an ideal of R, then S+I is a subring, I is an ideal of S+I, and S∩ I is an ideal of S

Theorem

Second isomorphism theorem for rings: S/(S∩ I)≅(S+I)/I

Lemma

If I⊆ J are ideals of R, then J/I is an ideal of R/I

Theorem

Third isomorphism theorem for rings: (R/I)/(J/I)≅ R/J

Theorem

Correspondence theorem: ideals of R/I correspond to ideals of R containing I

Definition

Prime ideals and maximal ideals in a commutative ring

Theorem

R/P is an integral domain if and only if P is a prime ideal

Theorem

R/M is a field if and only if M is a maximal ideal

Corollary

Every maximal ideal of a commutative ring is prime

Theorem

In a nonzero commutative ring, every proper ideal is contained in a maximal ideal

Sources