3.7 Rings

Definition of rings

A ring is a set equipped with two operations, addition and multiplication, that follow specific rules. Rings generalize familiar number systems like the integers and give you a precise framework for studying how addition and multiplication interact across many different mathematical settings.

Think of it this way: you already know groups, which capture the idea of one operation with nice properties. A ring layers a second operation on top and connects the two through the distributive law. This makes rings the natural structure for studying anything that involves both adding and multiplying.

Properties of rings

A ring is a set $R$ together with two binary operations, addition $(+)$ and multiplication $(\times)$, satisfying these axioms:

1. $(R, +)$ is an abelian group: addition is closed, associative, commutative, has an identity element $0$, and every element has an additive inverse.
2. Multiplication is associative: $(a \times b) \times c = a \times (b \times c)$ for all $a, b, c \in R$.
3. Distributive laws connect the two operations:
   • Left: $a \times (b + c) = (a \times b) + (a \times c)$
   • Right: $(b + c) \times a = (b \times a) + (c \times a)$

Notice what's not required: multiplication doesn't have to be commutative, and there doesn't have to be a multiplicative identity. Those are extra properties that define special types of rings.

Examples of common rings

• Integers $(\mathbb{Z})$ under standard addition and multiplication. This is the most natural example of a ring.
• Real numbers $(\mathbb{R})$ and complex numbers $(\mathbb{C})$ with their usual operations. These are actually fields (a stronger structure), but every field is a ring.
• Polynomial rings like $\mathbb{Z}[x]$, $\mathbb{R}[x]$, $\mathbb{C}[x]$. You add and multiply polynomials using the standard rules, and all the ring axioms hold.
• Matrix rings of $n \times n$ matrices over a ring. Matrix addition and multiplication satisfy the ring axioms, but note that matrix multiplication is not commutative.

Ring operations

Addition in rings

The additive structure of a ring is an abelian group $(R, +)$ with identity element $0$. This means addition satisfies closure, associativity, commutativity, and every element $a$ has an additive inverse $-a$.

Subtraction is defined through additive inverses: $a - b$ just means $a + (-b)$. Familiar examples include matrix addition and polynomial addition, both of which inherit this group structure.

Multiplication in rings

Multiplication in a ring must be closed and associative, but that's it for the base definition. Two things that might surprise you:

• Multiplication does not need to be commutative. In matrix rings, $A \times B \neq B \times A$ in general.
• There doesn't need to be a multiplicative identity. The set of even integers under standard operations forms a ring with no element that acts like $1$.

Distributive law

The distributive law is what ties addition and multiplication together. Without it, the two operations would just coexist without interacting.

• Left: $a \times (b + c) = (a \times b) + (a \times c)$
• Right: $(b + c) \times a = (b \times a) + (c \times a)$

You need both versions because multiplication might not be commutative. In a commutative ring, one implies the other, but in general you must verify both.

Types of rings

Commutative rings

A ring is commutative if $a \times b = b \times a$ for all $a, b \in R$. The integers, real numbers, and polynomial rings over commutative rings are all commutative.

Commutativity simplifies a lot of algebra. Many important structures, like integral domains and fields, are built on top of commutative rings.

Non-commutative rings

In a non-commutative ring, there exist elements $a, b$ where $a \times b \neq b \times a$. The order of multiplication matters, so you have to be careful when rearranging expressions.

The classic examples are matrix rings ($2 \times 2$ matrices over $\mathbb{R}$, for instance) and the quaternions. Non-commutative rings show up in physics (quantum mechanics) and computer graphics (rotations via quaternions).

Rings with unity

A ring with unity (or "unital ring") contains a multiplicative identity element $1$ satisfying $1 \times a = a \times 1 = a$ for all $a \in R$.

Most rings you'll encounter have unity: $\mathbb{Z}$, $\mathbb{R}$, polynomial rings, matrix rings. Having a $1$ lets you talk about units (elements with multiplicative inverses), which becomes important for studying fields and factorization.

Subrings

A subring is a subset of a ring that is itself a ring under the same operations. Subrings help you understand the internal structure of a ring by identifying smaller rings sitting inside it.

Definition of subrings

A subset $S$ of a ring $R$ is a subring if $S$ forms a ring using the same addition and multiplication from $R$. For example, the even integers $2\mathbb{Z}$ form a subring of $\mathbb{Z}$, and $\mathbb{R}$ is a subring of $\mathbb{C}$.

Criteria for subrings

To verify that $S$ is a subring of $R$, check three things:

1. Non-empty: $S \neq \emptyset$ (typically you show $0 \in S$).

2. Closed under subtraction: For all $a, b \in S$, $a - b \in S$.

3. Closed under multiplication: For all $a, b \in S$, $a \times b \in S$.

That's it. Closure under subtraction automatically gives you closure under addition, the additive identity, and additive inverses. If $R$ has unity and you want $S$ to be a subring with unity, you also need $1 \in S$.

Ring homomorphisms

A ring homomorphism is a function between two rings that preserves both operations. Homomorphisms let you compare rings and transfer structural information from one ring to another.

Definition of homomorphisms

A function $f: R \to S$ is a ring homomorphism if it satisfies:

1. $f(a + b) = f(a) + f(b)$ for all $a, b \in R$ (preserves addition)
2. $f(a \times b) = f(a) \times f(b)$ for all $a, b \in R$ (preserves multiplication)

If $R$ and $S$ are rings with unity, you typically also require $f(1_R) = 1_S$.

A concrete example: the inclusion map $\mathbb{Z} \hookrightarrow \mathbb{Q}$ that sends each integer to itself inside the rationals. It preserves both addition and multiplication.

Kernel and image

Every homomorphism $f: R \to S$ gives you two important subsets:

• Kernel: $\ker(f) = \{a \in R \mid f(a) = 0_S\}$. These are the elements that get "collapsed" to zero. The kernel is always an ideal of $R$ (not just a subring).
• Image: $\text{im}(f) = \{f(a) \mid a \in R\}$. This is the subring of $S$ that $f$ actually hits.

The kernel measures how far $f$ is from being injective. If $\ker(f) = \{0\}$, then $f$ is one-to-one.

Ideals

An ideal is a special kind of subring that "absorbs" multiplication by any element of the ring. Ideals are to rings what normal subgroups are to groups: they're exactly the subsets you can quotient by.

Left vs right ideals

• A left ideal $I$ satisfies: for all $r \in R$ and $a \in I$, $ra \in I$.
• A right ideal $I$ satisfies: for all $r \in R$ and $a \in I$, $ar \in I$.
• A two-sided ideal (usually just called an "ideal") is both a left and right ideal.

In commutative rings, the distinction disappears since $ra = ar$. The left/right distinction only matters in non-commutative rings.

Two-sided ideals

A two-sided ideal $I$ absorbs multiplication from both sides: $ra \in I$ and $ar \in I$ for all $r \in R$, $a \in I$. Two-sided ideals are used to construct quotient rings.

A key example: the even integers $2\mathbb{Z}$ form an ideal in $\mathbb{Z}$. Multiply any integer by an even integer and you get an even integer.

Principal ideals

A principal ideal is generated by a single element $a$:

$(a) = \{ra \mid r \in R\}$

In $\mathbb{Z}$, every ideal is principal. For instance, the ideal of all multiples of 3 is $(3) = 3\mathbb{Z}$. A ring where every ideal is principal is called a principal ideal domain (PID). The integers are the standard example of a PID.

Quotient rings

A quotient ring $R/I$ is formed by partitioning a ring $R$ into cosets of an ideal $I$, then defining operations on those cosets. This is the ring-theory analog of quotient groups.

Construction of quotient rings

Given a ring $R$ and an ideal $I$:

1. The elements of $R/I$ are cosets of the form $r + I$, where $r \in R$.
2. Addition: $(a + I) + (b + I) = (a + b) + I$
3. Multiplication: $(a + I)(b + I) = (ab) + I$
4. The natural projection $\pi: R \to R/I$ is defined by $\pi(r) = r + I$.

The ideal $I$ is what makes these operations well-defined. If $I$ were just a subring (not an ideal), multiplication of cosets wouldn't be consistent.

Properties of quotient rings

• $R/I$ is a ring with well-defined addition and multiplication.
• If $R$ is commutative, so is $R/I$.
• If $R$ has unity, then $1 + I$ is the unity of $R/I$.
• The kernel of the natural projection $\pi$ is exactly $I$.

A powerful example: $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ gives you a field with $p$ elements. This is how finite fields are constructed.

Special elements in rings

Units and invertible elements

A unit is an element $a \in R$ that has a multiplicative inverse: there exists $b \in R$ with $ab = ba = 1$. The set of all units in $R$, often written $R^*$ or $U(R)$, forms a group under multiplication.

In $\mathbb{Z}$, the only units are $1$ and $-1$. In a field, every non-zero element is a unit. In the ring of $2 \times 2$ real matrices, the units are exactly the invertible matrices (those with non-zero determinant).

Zero divisors

A zero divisor is a non-zero element $a$ for which there exists a non-zero $b$ with $ab = 0$. Zero divisors are "pathological" in the sense that they break the cancellation law.

For example, in $\mathbb{Z}/6\mathbb{Z}$: $2 \times 3 = 6 \equiv 0$, so $2$ and $3$ are both zero divisors. In matrix rings, any singular matrix (determinant zero) is a zero divisor.

A ring with no zero divisors (plus commutativity and unity) is an integral domain.

Nilpotent elements

An element $a \in R$ is nilpotent if $a^n = 0$ for some positive integer $n$. The smallest such $n$ is the nilpotency index.

Every nilpotent element (other than $0$ itself) is a zero divisor. To see why: if $a^n = 0$ and $n$ is the smallest such power, then $a \cdot a^{n-1} = 0$ with both $a \neq 0$ and $a^{n-1} \neq 0$.

A concrete example: strictly upper triangular matrices. In the ring of $3 \times 3$ matrices, a strictly upper triangular matrix $A$ satisfies $A^3 = 0$.

Integral domains

An integral domain is a commutative ring with unity that has no zero divisors. This is the right setting for studying factorization and divisibility, since the absence of zero divisors means you can cancel non-zero factors.

Definition and properties

The formal definition: a commutative ring $R$ with $1 \neq 0$ such that for all $a, b \in R$, if $ab = 0$ then $a = 0$ or $b = 0$.

The cancellation law follows directly: if $ab = ac$ and $a \neq 0$, then $a(b - c) = 0$, which forces $b - c = 0$, so $b = c$.

Every field is an integral domain (if $a \neq 0$ and $ab = 0$, multiply both sides by $a^{-1}$ to get $b = 0$). But not every integral domain is a field. The integers are an integral domain but not a field, since most integers lack multiplicative inverses.

Examples of integral domains

• Integers $(\mathbb{Z})$: the prototypical example.
• Polynomial rings over integral domains: $\mathbb{Z}[x]$, $\mathbb{R}[x]$.
• Gaussian integers $(\mathbb{Z}[i] = \{a + bi \mid a, b \in \mathbb{Z}\})$: these are used in number theory to study problems about sums of squares.
• Every integral domain can be embedded in its field of fractions. For $\mathbb{Z}$, this gives $\mathbb{Q}$. For $\mathbb{Z}[x]$, this gives the field of rational functions.

Fields

A field is a commutative ring with unity where every non-zero element has a multiplicative inverse. Fields are where you can do full arithmetic: add, subtract, multiply, and divide (by non-zero elements).

Definition of fields

Formally, a field $F$ is a commutative ring with $1 \neq 0$ such that for every non-zero $a \in F$, there exists $b \in F$ with $ab = 1$.

The familiar examples are $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. But there are also finite fields: $\mathbb{Z}/p\mathbb{Z}$ is a field whenever $p$ is prime.

Relationship to rings

Here's how the hierarchy works:

Each level adds a requirement:

• Commutative ring with unity: add commutativity of multiplication and a $1$.
• Integral domain: add the no-zero-divisors condition.
• Field: add multiplicative inverses for all non-zero elements.

You can sometimes build a field from a ring. The construction $\mathbb{Z}/p\mathbb{Z}$ for prime $p$ turns the ring of integers into a finite field. The field of fractions construction turns any integral domain into a field.

Ring theorems

Fundamental homomorphism theorem

Also called the First Isomorphism Theorem for rings: if $f: R \to S$ is a ring homomorphism, then

$R/\ker(f) \cong \text{im}(f)$

This says the quotient of $R$ by the kernel of $f$ is isomorphic to the image of $f$. It's the same idea as the group version, and it's one of the most-used tools in algebra. Whenever you have a homomorphism, this theorem lets you understand its image by studying a quotient ring instead.

Isomorphism theorems

• First Isomorphism Theorem: $R/\ker(f) \cong \text{im}(f)$ (same as above).
• Second Isomorphism Theorem: For a subring $S$ and ideal $I$ of $R$, $(S + I)/I \cong S/(S \cap I)$.
• Third Isomorphism Theorem: For ideals $I \subseteq J$ of $R$, $(R/I)/(J/I) \cong R/J$.

These theorems give you tools for simplifying quotient ring expressions and relating different quotient constructions to each other. They parallel the isomorphism theorems for groups.

Applications of ring theory

Cryptography

Ring structures underlie several modern cryptographic systems. Lattice-based cryptography relies on hard computational problems in polynomial rings, and the Ring Learning with Errors (Ring-LWE) problem is a foundation for post-quantum cryptography. Schemes like NTRU use polynomial rings over $\mathbb{Z}/q\mathbb{Z}$ for encryption.

Coding theory

Error-correcting codes frequently use polynomial rings over finite fields. Reed-Solomon codes, used in everything from CDs to deep-space communication, exploit properties of polynomial evaluation and interpolation over finite fields. Cyclic codes correspond directly to ideals in quotient polynomial rings.

Algebraic geometry

Algebraic geometry studies geometric objects (curves, surfaces, varieties) defined by polynomial equations. The connection to ring theory runs deep: points on a variety correspond to maximal ideals in a polynomial ring, and Hilbert's Nullstellensatz makes this correspondence precise. Commutative ring theory is the algebraic engine behind modern algebraic geometry.