Ideals and Quotient Rings: Examples and Counterexamples

Given: The ring $C$ of rational Cauchy sequences and its subset $N$ of null sequences.

Maximal ideals are maximal among proper ideals (Prime ideals and maximal ideals in a commutative ring).

Rational Cauchy sequences form a commutative ring $C$ (Cauchy sequences form a commutative ring).

$N$ is an ideal of $C$ (Null sequences form an ideal).

The null ideal $N$ is maximal in $C$ (The null ideal is maximal).

Given: The Cauchy-sequence ring $C$ and the null ideal $N$.

The real numbers are defined as the quotient of rational Cauchy sequences by null sequences (The real numbers).

Rational Cauchy sequences form the ring $C$ (Cauchy sequences form a commutative ring).

$N$ is an ideal of $C$ (Null sequences form an ideal).

$N$ is maximal in $C$ (The null ideal is maximal).

Quotienting a commutative ring by a maximal ideal gives a field ($R/M$ is a field if and only if $M$ is a maximal ideal).

The constructed real numbers form a field (The reals form a field).

Given: An integer $n>1$.

$\mathbb Z/n$ is the quotient ring $\mathbb Z/n\mathbb Z$ (For every $n\in\mathbb N$, the congruence-class ring $\mathbb Z/n$ is the quotient ring $\mathbb Z/n\mathbb Z$).

A quotient is a field exactly for a maximal ideal ($R/M$ is a field if and only if $M$ is a maximal ideal).

A maximal ideal in a commutative ring is prime (Every maximal ideal of a commutative ring is prime).

An integer greater than $1$ is prime exactly when it has the Euclid property (For an integer $p > 1$: $p$ is prime if and only if, for all integers $a$ and $b$, $p \mid ab$ implies $p \mid a$ or $p \mid b$).

$\mathbb Z/p$ is a field for prime $p$ (For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field).

$\mathbb Z$ is a commutative ring (The integers form a commutative ring).

Given: The zero ideal $(0)$ in $\mathbb Z$.

A prime ideal is proper and satisfies the zero-product implication; a maximal ideal has no proper intermediate ideal (Prime ideals and maximal ideals in a commutative ring).

The ideal criterion verifies subtraction closure and absorption (Ideal criteria and intersections of ideals).

$\mathbb Z$ is a nonzero commutative ring (The integers form a commutative ring).

Integer cancellation implies $ab=0$ only if $a=0$ or $b=0$ (The integers have no zero divisors; multiplicative cancellation).

Given: The subset $2\mathbb Z\subseteq\mathbb Z$.

A two-sided ideal is an additive subgroup with multiplication absorption (Left, right and two-sided ideals).

The ideal criterion is subtraction closure and absorption (Ideal criteria and intersections of ideals).

A subring contains the ambient identity (Subring: a subset containing $1_R$ and closed under addition, additive inverses and multiplication).

$\mathbb Z$ is a ring with identity $1$ (The integers form a commutative ring).

Given: Rings $R,S$ and the coordinate projection $p:R\times S\to S$, $p(r,s)=s$.

The product ring has coordinatewise operations (The product ring $R \times S$ with componentwise operations, its identity $(1_R, 1_S)$ and its units $R^{\times} \times S^{\times}$).

A ring homomorphism preserves operations and identity (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$).

A ring-homomorphism kernel is a two-sided ideal (The kernel of a ring homomorphism is a two-sided ideal).

The first ring isomorphism theorem gives quotient-by-kernel isomorphisms (First isomorphism theorem for rings: $R/\ker f\cong\operatorname{im}f$).

Ideals are additive subgroups with absorption (Left, right and two-sided ideals).