Identity and Inverse Elements | Malone Math

The Central Idea

An algebraic structure may contain an element that acts neutrally under its operation.

If \(e\) satisfies

\[ e\star a=a \]

and

\[ a\star e=a \]

for every \(a\) in the set, then \(e\) is an identity element.

Once an identity exists, we can ask whether each element has a partner that combines with it to produce that identity. Such a partner is called an inverse.

The Identity Element

Let \((S,\star)\) be a set equipped with a binary operation.

An element \(e\in S\) is an identity element if

\[ \boxed{ e\star a=a\star e=a } \]

for every

\[ a\in S. \]

The identity does not change any element when combined with it.

Left and Right Identities

It is useful to separate the two sides of the identity law.

A left identity satisfies

\[ e\star a=a \]

for every \(a\in S\).

A right identity satisfies

\[ a\star e=a \]

for every \(a\in S\).

A two-sided identity satisfies both conditions.

Why Both Sides Matter

In a commutative operation, left and right identities automatically agree because

\[ e\star a=a\star e. \]

But not every operation is commutative.

In noncommutative systems, an element might behave correctly on one side but not the other.

Therefore the standard identity definition requires both

\[ e\star a=a \]

and

\[ a\star e=a. \]

Additive Identity

Under ordinary addition, the identity element is

\[ \boxed{0}. \]

For every real number \(a\),

\[ 0+a=a \]

and

\[ a+0=a. \]

Therefore \(0\) is called the additive identity.

Multiplicative Identity

Under ordinary multiplication, the identity element is

\[ \boxed{1}. \]

For every real number \(a\),

\[ 1\cdot a=a \]

and

\[ a\cdot1=a. \]

Therefore \(1\) is called the multiplicative identity.

The Identity Depends on the Operation

The same set can have different identity elements under different operations.

On \(\mathbb{R}\),

\[ (\mathbb{R},+) \]

has identity

\[ 0, \]

while

\[ (\mathbb{R},\cdot) \]

has identity

\[ 1. \]

Identity is a property of the set-operation structure, not of the set alone.

Some Operations Have No Identity

Consider subtraction on \(\mathbb{Z}\).

A right identity would need to satisfy

\[ a-e=a \]

for every integer \(a\).

This gives

\[ e=0. \]

Indeed,

\[ a-0=a. \]

But \(0\) is not a left identity because

\[ 0-a=-a \]

is not generally equal to \(a\).

Therefore subtraction has no two-sided identity on \(\mathbb{Z}\).

Example: A Custom Operation with an Identity

On \(\mathbb{Z}\), define

\[ a\star b=a+b+1. \]

We seek \(e\) such that

\[ a\star e=a. \]

Using the rule,

\[ a+e+1=a. \]

Subtracting \(a\),

\[ e+1=0, \]

\[ e=-1. \]

Check the other side:

\[ (-1)\star a = -1+a+1 = a. \]

Therefore,

\[ \boxed{-1} \]

is the identity for \(\star\).

Finding an Identity Algebraically

For a formula-defined operation, a reliable method is:

Assume an identity element \(e\) exists.
Set \(a\star e=a\).
Solve for \(e\).
Check that the same \(e\) satisfies \(e\star a=a\).
Verify that \(e\) belongs to the underlying set.

Solving only one side is not enough unless commutativity has already been established.

The Identity Is Unique

If a binary operation has a two-sided identity, that identity is unique.

Suppose \(e\) and \(f\) are both identity elements.

Because \(e\) is an identity,

\[ e\star f=f. \]

Because \(f\) is an identity,

\[ e\star f=e. \]

Therefore,

\[ \boxed{e=f}. \]

There cannot be two distinct two-sided identity elements for the same operation.

Inverse Elements

Suppose \((S,\star)\) has identity element \(e\).

An element \(b\in S\) is an inverse of \(a\in S\) if

\[ \boxed{ a\star b=b\star a=e }. \]

The inverse combines with \(a\) to return the identity.

Inverse Means “Undo Relative to the Operation”

The word inverse depends on the operation being used.

Under addition, the inverse of \(a\) is the element that returns the additive identity:

\[ a+(-a)=0. \]

Under multiplication, the inverse of nonzero \(a\) is the element that returns the multiplicative identity:

\[ a\cdot\frac1a=1. \]

The relevant identity determines what it means to undo an element.

Additive Inverses

Under addition, the inverse of \(a\) is

\[ \boxed{-a}. \]

This is because

\[ a+(-a)=0 \]

and

\[ (-a)+a=0. \]

Examples include

\[ 7+(-7)=0 \]

and

\[ -12+12=0. \]

Multiplicative Inverses

Under multiplication, a nonzero number \(a\) has inverse

\[ \boxed{\frac1a}. \]

Since

\[ a\cdot\frac1a=1 \]

and

\[ \frac1a\cdot a=1, \]

\(\frac1a\) is called the multiplicative inverse or reciprocal of \(a\).

Zero Has No Multiplicative Inverse

A multiplicative inverse of \(0\) would need to be some number \(b\) satisfying

\[ 0\cdot b=1. \]

But

\[ 0\cdot b=0 \]

for every number \(b\).

Therefore no such \(b\) exists.

Thus

\[ \boxed{0\text{ has no multiplicative inverse}}. \]

Not Every Element Must Have an Inverse

The existence of an identity does not guarantee that every element has an inverse.

Consider

\[ (\mathbb{Z},\cdot). \]

The multiplicative identity is \(1\).

The integer \(2\) would need an integer \(b\) satisfying

\[ 2b=1. \]

That would require

\[ b=\frac12, \]

but

\[ \frac12\notin\mathbb{Z}. \]

Therefore \(2\) has no multiplicative inverse in \(\mathbb{Z}\).

Changing the Set Can Create Inverses

The number \(2\) has no multiplicative inverse in \(\mathbb{Z}\), but it does have one in \(\mathbb{Q}\):

\[ 2\cdot\frac12=1. \]

Thus inverse existence depends on the underlying set.

This is a major reason fields such as

\[ \mathbb{Q},\qquad\mathbb{R},\qquad\mathbb{C} \]

behave differently from the integers under multiplication.

Self-Inverse Elements

An element can be its own inverse.

Under addition,

\[ 0+0=0, \]

so \(0\) is its own additive inverse.

Under multiplication,

\[ 1\cdot1=1, \]

so \(1\) is its own multiplicative inverse.

Also,

\[ (-1)(-1)=1, \]

so \(-1\) is its own multiplicative inverse.

Left and Right Inverses

In noncommutative settings, it can be useful to distinguish two ideas.

A left inverse of \(a\) satisfies

\[ b\star a=e. \]

A right inverse satisfies

\[ a\star b=e. \]

A two-sided inverse satisfies both.

In group-like structures, these notions ultimately agree, but that result depends on additional structural properties.

Inverse Notation

In a general multiplicative-style structure, the inverse of \(a\) is often written

\[ \boxed{a^{-1}}. \]

This notation means

\[ a\star a^{-1} = a^{-1}\star a = e. \]

In an additive structure, the inverse is usually written

\[ -a. \]

The notation reflects the operation being used.

Do Not Confuse an Inverse with a Reciprocal

A reciprocal is specifically a multiplicative inverse of a nonzero number.

For example,

\[ \frac15 \]

is the reciprocal of \(5\) because

\[ 5\cdot\frac15=1. \]

But the word inverse is broader.

The additive inverse of \(5\) is

\[ -5, \]

not \(\frac15\).

Example: Inverses for a Custom Operation

On \(\mathbb{Z}\), let

\[ a\star b=a+b+1. \]

We already found that the identity is

\[ e=-1. \]

To find the inverse of \(a\), solve

\[ a\star b=-1. \]

Using the definition,

\[ a+b+1=-1. \]

Therefore,

\[ b=-a-2. \]

So the inverse of \(a\) under \(\star\) is

\[ \boxed{-a-2}. \]

Checking the Custom Inverse

Let

\[ b=-a-2. \]

Then

\[ a\star b = a+(-a-2)+1 = -1. \]

Also,

\[ b\star a = (-a-2)+a+1 = -1. \]

Since \(-1\) is the identity,

\[ \boxed{b=-a-2} \]

is indeed the inverse of \(a\).

Finding an Inverse Algebraically

Once the identity \(e\) is known:

Choose an arbitrary element \(a\in S\).
Let \(b\) be a possible inverse.
Solve \(a\star b=e\).
Check \(b\star a=e\).
Verify that \(b\in S\).

If no element of \(S\) satisfies the required equations, then \(a\) has no inverse in that structure.

Identity and Inverses in Function Composition

Consider functions from a set \(X\) to itself under composition.

The identity element is the identity function

\[ \operatorname{id}_X(x)=x. \]

For any function \(f:X\to X\),

\[ \operatorname{id}_X\circ f=f \]

and

\[ f\circ\operatorname{id}_X=f. \]

Therefore \(\operatorname{id}_X\) is the compositional identity.

Inverse Functions as Algebraic Inverses

A function \(f:X\to X\) has a two-sided inverse under composition when there exists a function \(f^{-1}:X\to X\) such that

\[ f^{-1}\circ f=\operatorname{id}_X \]

and

\[ f\circ f^{-1}=\operatorname{id}_X. \]

This is the familiar inverse-function idea expressed as an algebraic inverse under the operation of composition.

Identity Matrix

Under matrix multiplication, the role of \(1\) is played by the identity matrix.

For \(2\times2\) matrices,

\[ I= \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}. \]

For every \(2\times2\) matrix \(A\),

\[ IA=A \]

and

\[ AI=A. \]

Thus \(I\) is the multiplicative identity for square matrices of that size.

Inverse Matrices

If a square matrix \(A\) has an inverse matrix \(A^{-1}\), then

\[ AA^{-1}=I \]

and

\[ A^{-1}A=I. \]

Not every square matrix has an inverse.

Thus matrix multiplication gives another important example of a structure with an identity in which some elements may fail to be invertible.

Identity for Set Union

Consider the power set

\[ \mathcal{P}(U) \]

under union.

The empty set satisfies

\[ A\cup\varnothing=A \]

and

\[ \varnothing\cup A=A. \]

Therefore,

\[ \boxed{\varnothing} \]

is the identity element for union.

Identity for Set Intersection

On \(\mathcal{P}(U)\), intersection has a different identity.

For every \(A\subseteq U\),

\[ A\cap U=A \]

and

\[ U\cap A=A. \]

Therefore,

\[ \boxed{U} \]

is the identity for intersection.

Most Sets Do Not Have Inverses Under Union

The identity for union is

\[ \varnothing. \]

An inverse of \(A\) under union would need to be a set \(B\) satisfying

\[ A\cup B=\varnothing. \]

This is impossible unless

\[ A=\varnothing \]

and

\[ B=\varnothing. \]

Thus most elements of \(\mathcal{P}(U)\) have no inverse under union.

Identity Does Not Mean Inverse

The identity is one distinguished element of the whole structure.

An inverse is defined relative to a particular element.

Under addition:

\[ 0 \]

is the identity, while the inverse of \(7\) is

\[ -7. \]

Under multiplication:

\[ 1 \]

is the identity, while the inverse of \(7\) is

\[ \frac17. \]

Why Identity Comes Before Inverse

The definition of inverse depends on the identity.

To say that \(b\) is an inverse of \(a\), we must know which element the product

\[ a\star b \]

is supposed to equal.

Therefore the logical order is:

identify the operation;
find the identity;
then look for inverses.

Inverse Elements and Equation Solving

Inverses explain why familiar equation-solving procedures work.

\[ x+5=12, \]

adding the additive inverse \(-5\) gives

\[ x+5+(-5)=12+(-5). \]

Since

\[ 5+(-5)=0, \]

the identity law gives

\[ x+0=7, \]

\[ x=7. \]

Multiplicative Inverses and Equation Solving

\[ 5x=20, \]

multiply both sides by the inverse of \(5\):

\[ \frac15(5x)=\frac15(20). \]

Then

\[ \left(\frac15\cdot5\right)x=4. \]

Since

\[ \frac15\cdot5=1, \]

the identity law gives

\[ 1x=4, \]

\[ x=4. \]

Abstract algebra generalizes this familiar inverse-operation logic.

Cancellation and Inverses

In structures with suitable associativity and inverses, cancellation laws can be derived.

Suppose

\[ a\star b=a\star c. \]

If \(a\) has an inverse \(a^{-1}\), then applying \(a^{-1}\) to the left side of both expressions can remove \(a\).

This idea becomes formal in the study of groups.

Thus cancellation is closely connected to identities, inverses, and associativity.

The Inverse of an Inverse

If \(b\) is the inverse of \(a\), then \(a\) is also the inverse of \(b\).

Indeed, if

\[ a\star b=b\star a=e, \]

then the same equations show that \(a\) satisfies the inverse condition for \(b\).

In inverse notation,

\[ \boxed{(a^{-1})^{-1}=a}. \]

The Identity Is Its Own Inverse

Since the identity satisfies

\[ e\star e=e, \]

it combines with itself to produce the identity.

Therefore,

\[ \boxed{e^{-1}=e}. \]

This generalizes familiar facts such as

\[ -0=0 \]

and

\[ 1^{-1}=1. \]

Uniqueness of Inverses in an Associative Structure

In a structure with an associative operation and a two-sided identity, a two-sided inverse of an element is unique.

Suppose \(b\) and \(c\) are both inverses of \(a\).

Then

\[ b=b\star e. \]

Since \(a\star c=e\),

\[ b=b\star(a\star c). \]

By associativity,

\[ b=(b\star a)\star c. \]

Since \(b\star a=e\),

\[ b=e\star c=c. \]

Therefore,

\[ \boxed{b=c}. \]

Operation Tables and Identity Elements

On a finite set, an identity element can often be found directly from an operation table.

If \(e\) is the identity, then the row labeled \(e\) must reproduce the column headings, because

\[ e\star a=a. \]

The column labeled \(e\) must reproduce the row headings, because

\[ a\star e=a. \]

This gives a practical visual test.

Example: Identity in Addition Modulo 3

Consider addition modulo \(3\) on

\[ S=\{0,1,2\}. \]

\(\oplus\)	\(0\)	\(1\)	\(2\)
\(0\)	\(0\)	\(1\)	\(2\)
\(1\)	\(1\)	\(2\)	\(0\)
\(2\)	\(2\)	\(0\)	\(1\)

The row for \(0\) reproduces

\[ 0,1,2, \]

and the column for \(0\) does the same.

Therefore,

\[ \boxed{0} \]

is the identity.

Inverses in Addition Modulo 3

The identity is \(0\).

An inverse of an element \(a\) must combine with \(a\) to produce \(0\).

From the table,

\[ 0\oplus0=0, \]

so \(0\) is its own inverse.

Also,

\[ 1\oplus2=0 \]

and

\[ 2\oplus1=0. \]

Therefore \(1\) and \(2\) are inverses of one another.

Identity and Inverse Are Structural Properties

The formulas

\[ a+0=a \]

and

\[ a+(-a)=0 \]

are not merely isolated arithmetic facts.

They illustrate a general pattern:

\[ a\star e=a \]

and

\[ a\star a^{-1}=e. \]

Generalized algebra studies this pattern independently of the particular objects involved.

Common Error: Using the Wrong Identity

The identity depends on the operation.

Under addition,

\[ 0 \]

is the identity.

Under multiplication,

\[ 1 \]

is the identity.

Therefore the additive inverse of \(a\) must produce \(0\), while the multiplicative inverse must produce \(1\).

Common Error: Assuming Every Element Has an Inverse

The existence of an identity does not imply that every element is invertible.

\[ (\mathbb{Z},\cdot), \]

the identity \(1\) exists, but \(2\) has no multiplicative inverse in \(\mathbb{Z}\).

Inverse existence must be checked element by element or proved from the defining structure.

Common Error: Forgetting the Underlying Set

The equation

\[ 2b=1 \]

has solution

\[ b=\frac12. \]

But whether this gives an inverse depends on the set.

In \(\mathbb{Q}\), it does.

In \(\mathbb{Z}\), it does not.

Common Error: Checking Only One Side

In a general noncommutative setting, proving

\[ a\star b=e \]

does not automatically prove

\[ b\star a=e. \]

A two-sided inverse must satisfy both unless additional structure has already shown that one side forces the other.

Common Error: Treating \(a^{-1}\) as \(1/a\) Everywhere

In ordinary multiplication of nonzero real numbers,

\[ a^{-1}=\frac1a. \]

But in a general algebraic structure, \(a^{-1}\) means the inverse with respect to the operation being studied.

For a function, \(f^{-1}\) may mean an inverse function.

For a matrix, \(A^{-1}\) is an inverse matrix.

The symbol does not always mean numerical reciprocal.

A Reliable Procedure for Identity Elements

Identify the set \(S\) and operation \(\star\).
Assume an identity \(e\in S\) exists.
Solve \(a\star e=a\) for arbitrary \(a\).
Check \(e\star a=a\).
Verify that the resulting \(e\) lies in \(S\).
If no single element works for every \(a\), no two-sided identity exists.

A Reliable Procedure for Inverse Elements

Find the identity element \(e\) first.
Choose an arbitrary element \(a\in S\).
Let \(b\in S\) be a possible inverse.
Solve \(a\star b=e\).
Check \(b\star a=e\).
Verify that the resulting \(b\) belongs to \(S\).
Record whether every element has an inverse or only some do.

Example: Full Identity-and-Inverse Workflow

On \(\mathbb{R}\), define

\[ a\star b=a+b-ab. \]

Find the identity

Require

\[ a\star e=a. \]

Then

\[ a+e-ae=a. \]

Subtract \(a\):

\[ e-ae=0. \]

Factor:

\[ e(1-a)=0. \]

A single identity must work for every \(a\), so

\[ e=0. \]

Check:

\[ a\star0=a+0-a(0)=a \]

and

\[ 0\star a=0+a-0(a)=a. \]

Thus

\[ \boxed{e=0}. \]

Find the inverse of \(a\)

We require

\[ a\star b=0. \]

Thus

\[ a+b-ab=0. \]

Rearrange:

\[ b(1-a)=-a. \]

If \(a\ne1\), then

\[ b=\frac{-a}{1-a} = \frac{a}{a-1}. \]

Therefore, for \(a\ne1\),

\[ \boxed{a^{-1}=\frac{a}{a-1}} \]

with respect to \(\star\).

But if \(a=1\), then

\[ 1\star b=1+b-b=1, \]

which can never equal the identity \(0\).

So \(1\) has no inverse under this operation.

Practice 1: Additive Identity

State the identity element for

\[ (\mathbb{Z},+). \]

Practice 2: Multiplicative Identity

State the identity element for

\[ (\mathbb{Q},\cdot). \]

Practice 3: Additive Inverse

Find the additive inverse of

\[ -13. \]

Practice 4: Multiplicative Inverse

Find the multiplicative inverse of

\[ -\frac{5}{7}. \]

Practice 5: No Inverse in the Set

Does \(3\) have a multiplicative inverse in \(\mathbb{Z}\)? Explain.

Practice 6: Find a Custom Identity

On \(\mathbb{R}\), define

\[ a\star b=a+b-4. \]

Find the identity element.

Practice 7: Find a Custom Inverse

For

\[ a\star b=a+b-4, \]

find the inverse of \(a\).

Practice 8: Identity for Union

What is the identity element for union on

\[ \mathcal{P}(U)? \]

Practice 9: Identity for Intersection

What is the identity element for intersection on

\[ \mathcal{P}(U)? \]

Practice 10: Identify the Error

A student says:

“The inverse of \(8\) is always \(\frac18\).”

Explain why this statement is incomplete.

Practice Answers

Practice 1

The additive identity is

\[ \boxed{0}. \]

Practice 2

The multiplicative identity is

\[ \boxed{1}. \]

Practice 3

The additive inverse of \(-13\) is

\[ \boxed{13}, \]

because

\[ -13+13=0. \]

Practice 4

The reciprocal of

\[ -\frac57 \]

\[ \boxed{-\frac75}. \]

Indeed,

\[ \left(-\frac57\right)\left(-\frac75\right)=1. \]

Practice 5

No. A multiplicative inverse \(b\in\mathbb{Z}\) would satisfy

\[ 3b=1, \]

\[ b=\frac13. \]

Since

\[ \frac13\notin\mathbb{Z}, \]

\(3\) has no multiplicative inverse in \(\mathbb{Z}\).

Practice 6

Require

\[ a\star e=a. \]

Then

\[ a+e-4=a, \]

\[ e=4. \]

Check:

\[ 4\star a=4+a-4=a. \]

Therefore,

\[ \boxed{e=4}. \]

Practice 7

The identity is \(4\). Require

\[ a\star b=4. \]

Thus

\[ a+b-4=4, \]

\[ b=8-a. \]

Therefore the inverse of \(a\) is

\[ \boxed{8-a}. \]

Practice 8

The identity for union is

\[ \boxed{\varnothing}, \]

because

\[ A\cup\varnothing=A. \]

Practice 9

The identity for intersection is

\[ \boxed{U}, \]

because

\[ A\cap U=A. \]

Practice 10

The inverse depends on the operation.

Under multiplication, the inverse of \(8\) is

\[ \frac18. \]

Under addition, the inverse of \(8\) is

\[ -8. \]

Therefore the operation must be specified.

What to Remember

An identity element \(e\) satisfies

\[ \boxed{ e\star a=a\star e=a }. \]

An inverse \(a^{-1}\) of \(a\) satisfies

\[ \boxed{ a\star a^{-1} = a^{-1}\star a = e }. \]

Key examples include:

\(0\) as the additive identity;
\(-a\) as the additive inverse of \(a\);
\(1\) as the multiplicative identity;
\(\frac1a\) as the multiplicative inverse of nonzero \(a\);
the identity function under composition;
the identity matrix under matrix multiplication.

The identity depends on the operation, and inverse existence depends on both the operation and the underlying set.

Where We Go Next

Identity and inverse elements become especially concrete when operations are displayed on finite sets.

The next lesson, Operation Tables, develops systematic methods for reading and constructing tables, checking closure and commutativity, locating identity elements, identifying inverses, and investigating finite algebraic structures.