Basic Algebraic Structures

Mar 12, 2026 - ⧖ 7.0 min

Algebraic Structures

Once we have defined our set we want to define the operations, normally binary operations, on the elements of such set. The operations we define must also satisfy a finite set of identities called axioms.

The combination of a nonempty set, the operations on the elements and the axioms for these operation is what we call algebric structure.

Binary operations

A binary operation on a set $S$ is a function

$$ \cdot : S \times S \to S $$

that takes two elements of $S$ and returns another element of $S$.

Semigroups

A semigroup is a pair $(S, \cdot)$ where $S$ is a set and $\cdot$ is a binary associative operation:

$$ \forall a, b, c \in S \implies (a \cdot b) \cdot c = a \cdot (b \cdot c) $$

Associativity is the key structural property here.

For brevity, we will often omit the symbol $\cdot$ and write $ab$ instead of $a \cdot b$, whenever the meaning is clear from the context.

Two elements $a, b \in S$ are said to commute if $ab = ba$. A semigroup is called abelian (or commutative semigroup) if every $a, b \in S$ commute, such that:

$$ \forall a, b \in S, \quad ab = ba $$

The term abelian is more commonly used in the context of groups, but the notion of commutativity applies to any algebraic structure with a binary operation.

We can also use the addition notation $+$, with it the associative property can be defined as the following:

$$ \forall a, b, c \in S \implies (a + b) + c = a + (b + c) $$

Cancellation Laws

Given a semigroup $(S, \cdot)$ and an element $x \in S$, we say that:

$x$ is left cancellable if, $\forall a, b \in S$, $$ xa = xb \implies a = b $$
$x$ is right cancellable if, $\forall a, b \in S$, $$ ax = bx \implies a = b $$

A semigroup in which every element is left and right cancellable is called a cancellative semigroup.

Not all semigroups satisfy the cancellation law. This is a strong additional property.

Powers in a Semigroup

Let $a \in S$, we define the positive powers of an element $a$ in the following way:

$$ a^{n} = \begin{cases} a^1 = a \ a^{n+1} = a^na \end{cases} $$

Because of associativity, expressions like $a \cdot a \cdot a$ do not depend on how we place parentheses, so $a^n$ is well-defined.

This is one of the first concrete consequences of associativity.

An element $x \in S$ is called idempotent if $x^2 = x$.

Example: In $(\mathbb{N}, +)$, the element $0$ is idempotent since $0 + 0 = 0$.

With the addition notation $+$ the powers are called multiplies, defined as

$$ a^{n} = \begin{cases} a^1 = a \ a^{n+1} = a^n + a \end{cases} $$

Monoids

A semigroup $(M, \cdot)$ is called a monoid if there exists $e \in M$ (called the identity element) such that

$$ \forall a \in M \quad ea = ae = a $$

This element is unique, indeed let $e, e' \in M$ two identity elements, then

$$ e = ee' = e' $$

Because of this, we can denote a monoid by the triple $(M, \cdot, e)$.

Powers in a Monoid

The existence of the identity allows us to extend the definition of powers. For every $a \in M$, we define: $$ a^0 = e $$

Together with the recursive definition $$ a^{n+1} = a^na $$ this gives a consistent definition of $a^n$ for all $n \in \mathbb{N}$.

The identity element is idempotent, indeed

$$ e^2 = ee = e $$

Invertible elements

Let $(M, \cdot, e)$ be a monoid, an element $a \in M$ is called invertible if there exists an element $b \in M$ such that

$$ ab = ba = e $$

The element $b$ is called the inverse of a, and it is unique, indeed let $b, c \in M$ be two inverse of $a$, then

$$ b = be = b(ac) = (ba)c = ec = c $$

The uniqueness of the inverse of an element $a$ allow us to denote it by $a^{-1}$.

The name inverse is used withing the multiplication notation, if we use the addition notation, then the inverse element is called the opposite

Groups

A group is a monoid $(G, \cdot, e)$ where every element has an inverse.

Formally, a group $(G, \cdot, e)$ satisfies:

Associativity of the $\cdot$ operation
There exists the identity element $e$
For every $a \in G$, there exists $a^{-1} \in G$ such that $$ aa^{-1} = a^{-1}a = e $$

If the operation is also commutative, the group is called abelian, of course a group is abelian if the underlying semigroup is commutative.

We can now demonstrate that every group satisfies the cancellation law, indeed if $ab = ac$ then

$$ b = eb = (a^{-1}a)b = a^{-1}(ab) = a^{-1}(ac) = (a^{-1}a)c = ec = c $$

similarly, if $ba = ca$ then

$$ b = be = b(aa^{-1}) = (ba)a^{-1} = (ca)a^{-1} = c(aa^{-1}) = ce = c $$

Rings

A ring is a triple $(R, +, \cdot)$ where $R$ is a set, $+$ is a binary operation and $\cdot$ is an associative binary operation such that the following properties hold:

$(R, +, 0)$ is an abelian group
$(R, \cdot)$ is a semigroup
Distributive laws hold: $$ a(b + c) = ab + ac $$ $$ (a + b)c = ac + bc $$

Given two elements $a, b \in R$ we say that they commute if $ab = ba$. A ring is called commutative ring if $\forall a, b \in R$, $ab = ba$.

Unitary rings

The $\cdot$ operation in our ring $(R, + \cdot)$ may not have an identity element 1 such that $\forall a \in R$, $a \cdot e = e \cdot a = a$, if this element does exist, $e$ is called the unity of the ring.

Like for groups, this element is unique and with it, the subgroup $(R, \cdot)$ becomes a monoid $(R, \cdot, 1)$, we can then call the ring $(R, +, \cdot)$ a unitary ring.

Commutative rings

Given a ring $(R, +, \cdot)$ we already know that the group under the $+$ operation is abelian, such that $\forall a, b \in R$, $a + b = b + a$.

We did not put such constraint for the $\cdot$ operation, if such property holds, thus $\forall a, b \in R$, $a \cdot b = b \cdot a$ we call the ring $(R, +, \cdot)$ a commutative ring.

Integral domains

To define an integral domain we must first take a step back and define what a zero divisor is. A zero divisor is an element $a \ne 0 \in R$ such that there exists $x \ne 0 \in R$ with $ax = 0$.

The basic set of numbers we can think of is the integers $\mathbb{Z}$ which does not include zero divisor, indeed $\forall a, b \in \mathbb{Z}, ab = 0 \iff a = 0 \text{ or } b = 0$.

We can see an example in $\mathbb{Z}/4\mathbb{Z}$ through a Cayley table:

·	1	2	3
0	0	0	0
1	1	2	3
2	2	0	2
3	3	2	1

We note that $2 \cdot 2 = 0$ so $2$ is a zero divisor, with it we can now finally define an integral domain.

An integral domain is a commutative ring $(R, +, \cdot)$ without zero divisors, like $\mathbb{Z}$.

Division rings

A division ring is a unitary ring $(R, +, \cdot)$ such that every non-zero element in $R$ have an inverse under the $\cdot$ operation, meaning that $(R - \{0\}, \cdot, 1)$ is a group.

The commutative property is not required, if it is present, then the structure becomes a field.

Fields

A field is a commutative unitary ring $(R, +, \cdot)$ such that every non-zero element in $R$ have an inverse under the $\cdot$ operation, meaning that $(R - \{0\}, \cdot, 1)$ is an abelian group.

Typo?