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?
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.
Contents
Existence and Properties
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?
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. \(_\square\)
Suppose \(S\) is a set with a binary operation. Consider the following sentence about the identity elements in \(S\):
\(S\) has \(\underline{\phantom{1234567}}\) left identities, \(\underline{\phantom{1234567}}\) right identities, and \(\underline{\phantom{1234567}}\) identity elements.
Which choice of words for the blanks gives a sentence that cannot be true?
Examples
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 \(a\in \mathbb R.\)