Identity Element

An identity element in a set is an element that is special with respect to a binary operation on the set: when an identity element is paired with any element via the operation, it returns that element.

More explicitly, let $S$ be a set and $*$ be a binary operation on $S.$ Then

• an element $e\in S$ is a left identity if $e*s = s$ for any $s \in S;$
• an element $f\in S$ is a right identity if $s*f = s$ for any $s \in S;$
• an element that is both a left and right identity is called a two-sided identity, or identity element, or identity for short.

Let $S = \{a,b,c,d\},$ and consider the binary operation defined by the following table: $$\begin{array}{|c|cccc|}\hline *&a&b&c&d \\ \hline a&a&a&a&a \\ b&c&b&d&b \\ c&d&c&b&c \\ d&a&b&c&d \\ \hline \end{array}$$ The value of $x * y$ is given by looking up the row with $x$ and the column with $y.$

What are the left identities? What are the right identities?

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The unique left identity is $d.$ The unique right identity is also $d.$ This is because the row corresponding to a left identity should read $a,b,c,d,$ as should the column corresponding to a right identity. Note that $*$ is not a commutative operation ($x*y$ and $y*x$ are not necessarily the same), so a left identity is not automatically a right identity (imagine the same table with the top right entry changed from $a$ to something else). $_\square$

Let $S= \mathbb R,$ the set of real numbers, and let $*$ be addition. Then $0$ is an identity element: $0+s = s+0 = s$ for any $s \in \mathbb R.$ Note that $0$ is the unique left identity, right identity, and identity element in this case.

In general, there may be more than one left identity or right identity; there also might be none.

Let $S = \mathbb R,$ and define $*$ by the formula $$a*b = a^2-3a+2+b.$$ What are the left identities, right identities, and identity elements?

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If $e$ is a left identity, then $e*b=b$ for all $b\in \mathbb R,$ so $e^2-3e+2+b=b,$ so $e^2-3e+2=0.$ This has two solutions, $e=1,2,$ so $1$ and $2$ are both left identities.

If $f$ is a right identity, then $a*f=a$ for all $a \in \mathbb R,$ so $a = a^2-3a+2+f,$ so $f = -a^2+4a-2.$ But no $f$ can be equal to $-a^2+4a-2$ for all $a \in \mathbb R$: for instance, taking $a=0$ gives $f=-2,$ but taking $a=1$ gives $f=1.$ This is impossible. So there are no right identities.

Because there is no element which is both a left and right identity, there is no identity element. $_\square$

It is the case that if an identity element exists, it is unique:

If $S$ is a set with a binary operation, and $e$ is a left identity and $f$ is a right identity, then $e=f$ and there is a unique left identity, right identity, and identity element.

By the properties of identities, $$e = e*f = f.$$ If $e'$ is another left identity, then $e'=f$ by the same argument, so $e'=e.$ So the left identity is unique. A similar argument shows that the right identity is unique. Since $e=f,$ it is both a left and a right identity, so it is an identity element, and any other identity element must equal it, by the same argument. $_\square$

• The set of subsets of $\mathbb Z$ (or any set) has a binary operation given by union. The identity for this operation is the empty set $\varnothing,$ since $\varnothing \cup A = A.$

• The set of subsets of $\mathbb Z$ (or any set) has another binary operation given by intersection. The identity for this operation is the whole set $\mathbb Z,$ since ${\mathbb Z} \cap A = A.$

• Every group has a unique two-sided identity element $e.$

• Every ring has two identities, the additive identity and the multiplicative identity, corresponding to the two operations in the ring. For instance, $\mathbb R$ is a ring with additive identity $0$ and multiplicative identity $1,$ since $0+a=a+0=a,$ and $1 \cdot a = a \cdot 1 = a$ for all $a\in \mathbb R.$