Rings, Ideals, and Quotient Rings

Part 11, Chapter 11: Abstract Algebra Survey

Learning objectives

State the ring axioms and distinguish commutative rings, rings with unity, and integral domains
Identify ideals (left, right, two-sided) and construct quotient rings
Recognise principal ideal domains (PIDs) and unique factorisation domains (UFDs)
Use the correspondence between ideals and quotient rings to compute with $\mathbb{Z}/n\mathbb{Z}$ and $k[x]/(p(x))$

A ring is what you get when you upgrade a group by adding multiplication. The integers, the polynomials, the matrices, and the continuous functions on an interval are all rings, they have addition (forming an abelian group) and a separate multiplication that distributes over addition. Once two operations are in play, the right notion of substructure changes: subgroups give way to ideals, and quotient groups give way to quotient rings. This is the algebraic machinery behind modular arithmetic, polynomial factorisation, and the construction of finite fields used in coding theory.

The ring axioms

A ring is a triple $(R, +, \cdot)$ where:

$(R, +)$ is an abelian group with identity $0$ .
$\cdot : R \times R \to R$ is associative: $(ab)c = a(bc)$ .
Distributivity: $a(b + c) = ab + ac$ and $(a + b)c = ac + bc$ for all $a, b, c \in R$ .

A ring is commutative if $ab = ba$ . It has unity if there is $1 \in R$ with $1 \cdot a = a \cdot 1 = a$ for every $a$ . Most rings we will meet (and all rings in this section) have a unity and are commutative unless stated otherwise.

Examples to know cold

$\mathbb{Z}$ : commutative ring with unity. $\mathbb{Z}/n\mathbb{Z}$ : finite commutative ring with unity, a field exactly when $n$ is prime. $k[x]$ for a field $k$ : the polynomial ring in one variable, a Euclidean domain (you can do long division). $M_n(\mathbb{R})$ n(mathbbR): the ring of $n \times n$ real matrices, non-commutative for $n \geq 2$ . $C[0, 1]$ : continuous real-valued functions on $[0, 1]$ under pointwise operations, commutative, with unity the constant function $1$ .

The mod-clock above is now serving double duty: as a group it visualises $(\mathbb{Z}/n\mathbb{Z}, +)$ , and as a ring it visualises $(\mathbb{Z}/n\mathbb{Z}, +, \times)$ with multiplication implemented as "repeated stepping." Pick $n = 6$ and compute $3 \times 4$ mod $6$ : take $4$ stepped $3$ times, landing on $0$ . The fact that $3 \cdot 4 \equiv 0 \pmod 6$ with neither factor zero is your first encounter with zero divisors, the obstruction that prevents $\mathbb{Z}/n\mathbb{Z}$ from being a field when $n$ is composite.

Ideals: the right kind of subset

A subset $I \subseteq R$ is a (two-sided) ideal if:

$(I, +)$ is a subgroup of $(R, +)$ .
Absorption: for every $r \in R$ and $a \in I$ , both $ra \in I$ and $ar \in I$ .

Ideals are to rings what normal subgroups are to groups. The key fact: if $I$ is an ideal then the quotient $R / I$ inherits a well-defined ring structure with operations $(r + I) + (s + I) = (r + s) + I$ and $(r + I)(s + I) = rs + I$ . The most famous example is $\mathbb{Z}/n\mathbb{Z} = \mathbb{Z}/(n)$ , the quotient of $\mathbb{Z}$ by the ideal $(n) = n\mathbb{Z}$ of all multiples of $n$ .

Principal ideals, integral domains, UFDs

A principal ideal is one generated by a single element: $(a) = \{ra : r \in R\}$ . A commutative ring with unity in which every ideal is principal is a principal ideal domain (PID). The integers form a PID; so does $k[x]$ for any field $k$ .

An integral domain is a commutative ring with unity and no zero divisors: if $ab = 0$ then $a = 0$ or $b = 0$ . The integers are an integral domain; $\mathbb{Z}/6\mathbb{Z}$ is not (because $2 \cdot 3 = 0$ ). A unique factorisation domain (UFD) is an integral domain in which every nonzero non-unit factors uniquely (up to order and units) into irreducibles. Every PID is a UFD, the deep generalisation of the fundamental theorem of arithmetic to general rings.

Where this shows up

Error-correcting codes: Reed-Solomon, BCH, and Reed-Muller codes, the codes that keep CDs, QR codes, deep-space probes, and 5G working, are built inside the polynomial ring $\mathbb{F}_q[x]$ and its quotients. A codeword is an element of an ideal, and decoding reduces to polynomial arithmetic.
Cryptography: Lattice-based post-quantum schemes (Kyber, Dilithium, NTRU) live inside polynomial rings of the form $\mathbb{Z}[x] / (x^n + 1)$ . Security reduces to hard problems about ideals in those quotient rings.
Algebraic geometry: Hilbert's Nullstellensatz says that radical ideals in $\mathbb{C}[x_1, \ldots, x_n]$ correspond bijectively to algebraic varieties in $\mathbb{C}^n$ . Modern algebraic geometry is the study of rings up to the equivalence "they cut out the same shape."
Number theory: The ring of integers $\mathcal{O}_K$ K in a number field $K$ may fail to be a UFD, but its ideals always factor uniquely, the genesis of class field theory and the modern proof of Fermat's Last Theorem.

Pause and think: What is $(2) \cap (3)$ inside $\mathbb{Z}$ ? (Hint: this is the set of integers that are both multiples of $2$ and multiples of $3$ .) What single ideal is this equal to?

Try it

Predict first: list every ideal of $\mathbb{Z}/8\mathbb{Z}$ . (Hint: ideals of $\mathbb{Z}/n\mathbb{Z}$ correspond to divisors of $n$ .)
Find all zero divisors in $\mathbb{Z}/12\mathbb{Z}$ . Then list the units (elements with multiplicative inverses). What is the relationship between zero divisors and units? (Spoiler: in a finite commutative ring with unity, every nonzero element is either a unit or a zero divisor.)
Show that $\mathbb{Z}[x] / (x) \cong \mathbb{Z}$ . (Hint: in the quotient, $x = 0$ , so every polynomial collapses to its constant term.)
Factor $x^2 - 4$ inside $\mathbb{Z}[x]$ into irreducibles. Then factor $x^4 + 1$ inside $\mathbb{R}[x]$ , you will need a quadratic formula or completing the square.
True or false: $\mathbb{Z}[x]$ is a PID. (Hint: consider the ideal $(2, x)$ , is it principal?)

A trap to watch for

The single most common error is treating subring and ideal as synonyms. They are not. A subring is closed under $+, \cdot, -,$ and contains $1$ , it is a ring in its own right. An ideal is closed under $+, -$ and is absorbed by multiplication by any element of the larger ring. Most ideals are not subrings (they typically do not contain $1$ ), and most subrings are not ideals (they are not absorbed by outside multiplication). The integers $\mathbb{Z} \subseteq \mathbb{Q}$ are a subring but not an ideal: multiply $1 \in \mathbb{Z}$ by $1/2 \in \mathbb{Q}$ and you exit $\mathbb{Z}$ . The even integers $2\mathbb{Z} \subseteq \mathbb{Z}$ are an ideal but not a subring with unity: $1 \notin 2\mathbb{Z}$ .

What you now know

You can state the ring axioms, recognise standard examples, identify ideals and form quotient rings, and tell which finite quotients $\mathbb{Z}/n\mathbb{Z}$ are fields. The next section pushes one step further, to fields, where division is always possible, and to Galois theory, which turns the question "is this polynomial solvable by radicals?" into a question about finite groups.

Mark section complete →

References

Garrity, T. (2002). All the Mathematics You Missed: But Need to Know for Graduate School. Cambridge University Press, ch. 11.
Dummit, D. S., Foote, R. M. (2003). Abstract Algebra (3rd ed.). Wiley, ch. 7-9.
Artin, M. (2010). Algebra (2nd ed.). Pearson, ch. 11-12.
Lang, S. (2002). Algebra (3rd revised ed.). Springer, ch. 2-3.
Herstein, I. N. (1996). Abstract Algebra (3rd ed.). Wiley, ch. 4.