Identity Element¶

Definition¶

An identity element for a binary operation $\star$ on $A$ is an element $e\in A$ such that

for all $a\in A$$A$.

Interpretation¶

The identity element leaves every element unchanged under the operation: combining it with any $a$, on either side, gives $a$ back. Which element this is depends on the operation: $0$ for addition, $1$ for multiplication, and the function $x\mapsto x$ for composition of functions from a set to itself.

Both sides matter. For subtraction on $\mathbb{Z}$, $a-0=a$ but $0-a=-a$$-a$, so $0$ works only on the right and subtraction has no identity element. Addition on $\{1,2,3,\ldots\}$ has none on either side.

If an identity exists, it is unique: for identities $e$ and $e'$, $e\star e'=e$ because $e'$ is an identity and $e\star e'=e'$ because $e$ is one, so $e=e'$$e^{\prime }$. This is why one speaks of the identity and can define inverse elements with respect to it; the identity matrix plays this role later for matrix multiplication.

Key Results¶

• If an identity element exists, it is unique.
• Associativity, identity elements, and inverse elements determine which algebraic structure is present.

Examples¶

• $0$ is the identity for addition; $1$ is the identity for multiplication.

Prerequisites¶

• Binary Operation

Related Concepts¶

• Identity Matrix

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: neutral element, unit element Symbols: element of, integers

Last updated: 20 October 2026 03:02