Ideals and Quotient Rings

A subset $I \subseteq R$ is an ideal if:

1. $(I, +)$ is a subgroup of $(R, +)$.
2. For all $r \in R$ and $a \in I$: $ra \in I$ and $ar \in I$.

If only $ra \in I$ for all $r \in R$ and $a \in I$ Then $I$ is a left ideal. Similarly for right Ideals. A two-sided ideal (or ideal) satisfies both.

Proposition 9.1. Every ideal is a subring. The converse is false.

Example. $n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$ is an ideal of $\mathbb{Z}$.

Example. If $\phi : R \to S$ is a ring homomorphism, then $\ker(\phi)$ is an ideal of $R$.

If $I$ is an ideal of $R$ The quotient ring $R/I$ has elements $\{a + I : a \in R\}$ (cosets) With operations $(a + I) + (b + I) = (a + b) + I$ and $(a + I)(b + I) = ab + I$.

Theorem 9.2 (Ring Isomorphism Theorem). If $\phi : R \to S$ is a surjective ring homomorphism, Then $R / \ker(\phi) \cong S$.

The …/1-number-and-algebra/3_proof-and-logic follows the same pattern as the first isomorphism theorem for groups.

An ideal $I \neq R$ is prime if $ab \in I$ implies $a \in I$ or $b \in I$.

An ideal $I \neq R$ is maximal if there is no ideal $J$ with $I \subsetneq J \subsetneq R$.

Theorem 9.3. In a commutative ring $R$ with unity:

1. $I$ is a prime ideal if and only if $R/I$ is an integral domain.
2. $I$ is a maximal ideal if and only if $R/I$ is a field.

Corollary 9.4. Every maximal ideal is prime.

Proof. A field is an integral domain. $\blacksquare$

Example. In $\mathbb{Z}$ The ideal $(p)$ is maximal (hence prime) if and only if $p$ is prime. The ideal $(6)$ is neither prime nor maximal. The ideal $(0)$ is prime ($\mathbb{Z}$ is an integral domain) But not maximal ($\mathbb{Z}/(0) \cong \mathbb{Z}$ is not a field).

Example. In $\mathbb{R}[x]$ The ideal $(x^2 + 1)$ is maximal since $\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}$ is a field.

Problem. Show that $(2)$ is a maximal ideal of $\mathbb{Z}$ but $(4)$ is not.

Theorem 9.5 (Chinese Remainder Theorem for Rings). Let $R$ be a commutative ring with unity and Let $I, J$ be ideals with $I + J = R$. Then

Proof. Define $\phi : R \to R/I \times R/J$ by $\phi(r) = (r + I, r + J)$. This is a ring homomorphism. It is surjective: since $I + J = R$ There exist $a \in I$ and $b \in J$ with $a + b = 1$. For any $(r_1 + I, r_2 + J)$ Take $r = r_1b + r_2a$. Then $r \equiv r_1b \equiv r_1(1-a) \equiv r_1 \pmod{I}$ And $r \equiv r_2a \equiv r_2(1-b) \equiv r_2 \pmod{J}$.

The kernel is $\ker(\phi) = \{r : r \in I\ \mathrm{and\ r} \in J\} = I \cap J$. By the ring isomorphism theorem, $R/(I \cap J) \cong R/I \times R/J$. $\blacksquare$

Corollary 9.6. If $m, n \in \mathbb{Z}$ are coprime, then $\mathbb{Z}/(mn) \cong \mathbb{Z}/(m) \times \mathbb{Z}/(n)$.

Proof. Apply Theorem 9.5 with $I = (m)$, $J = (n)$. Since $\gcd(m, n) = 1$ We have $(m) + (n) = (1) = \mathbb{Z}$. Also $(m) \cap (n) = (\mathrm{lcm}(m, n)) = (mn)$. $\blacksquare$

Problem. Find all solutions to $x \equiv 2 \pmod{3}$, $x \equiv 3 \pmod{5}$, $x \equiv 1 \pmod{7}$.

An ideal is the ring-theoretic analogue of a normal subgroup. It is a subset that absorbs multiplication from both sides: if $a \in I$ and $r \in R$, then $ra$ and $ar$ are in $I$. This absorption property is what makes quotient rings possible: it ensures that the product of two cosets $(a + I)(b + I) = ab + I$ is well-defined.

Prime ideals are the ring-theoretic analogue of prime numbers: $I$ is prime if $ab \in I$ implies $a \in I$ or $b \in I$. The quotient by a prime ideal is an integral domain (no zero divisors). Maximal ideals are the largest proper ideals: the quotient by a maximal ideal is a field (every non-zero element is invertible). In $\mathbb{Z}$, the prime ideals are $(p)$ for each prime $p$, and these are also maximal, giving the fields $\mathbb{Z}/p\mathbb{Z}$. The Chinese Remainder Theorem says that when two ideals are coprime, the quotient by their product is isomorphic to the product of the individual quotients, which is the algebraic foundation for modular arithmetic and RSA cryptography.

• Confusing subrings with ideals. Every ideal is a subring, but subrings need not be closed under multiplication by arbitrary ring elements.
• Forgetting two-sided closure. An ideal must absorb multiplication from both sides: $ra \in I$ and $ar \in I$ for all $r \in R$, $a \in I$. In non-commutative rings, left and right ideals differ.
• Assuming primality implies maximality. In $\mathbb{Z}$, $(0)$ is prime but not maximal. In $\mathbb{R}[x,y]$, $(x)$ is prime but not maximal since $(x) \subsetneq (x,y)$.
• Misapplying CRT. The Chinese Remainder Theorem requires coprime moduli (or more generally, $I + J = R$). Without this condition, the natural map need not be surjective.

• Cryptography: RSA encryption relies on $\mathbb{Z}/(n)$ where $n = pq$; the CRT optimises decryption via the isomorphism $\mathbb{Z}/(pq) \cong \mathbb{Z}/(p) \times \mathbb{Z}/(q)$.
• Error-correcting codes: Polynomial rings over finite fields and quotient constructions underpin Reed-Solomon and BCH codes.
• Algebraic geometry: The correspondence between ideals of $k[x_1, \ldots, x_n]$ and algebraic varieties (Hilbert’s Nullstellensatz) generalises the prime/maximal ideal classification.

• Rings: Rings provide the ambient structure in which ideals and quotient rings are defined.

• Homomorphisms and Isomorphism Theorems: The kernel of a ring homomorphism is always an ideal, enabling the ring isomorphism theorem.

• Field Theory: Quotienting a polynomial ring by an irreducible ideal produces a field extension.

• Classical Mechanics

• Electromagnetism

• Additional Results
• Classification of Groups of Small Order
• Common Pitfalls
• Euclidean Domains, PIDs, and UFDs