In the monoid (Z,+,0) every element has an inverse: The inverse of a is -a.

In the monoid (Z,·,1) only the elements 1 and -1 have an inverse; they are their own inverses.

In the monoid (Z/10Z, ·, 1) only the elements 1, 3, 7, 9 have an inverse: 3 · 7 = 1 (mod 10) and 9 · 9 = 1 (mod 10).

Thanks to Cramers' rule we know that exactly those n by n matrices have an inverse with respect to matrix multiplication that have a nonzero determinant.

In the monoid M_n of all maps {1,...,n} -> {1,...,n}, in which multiplication is composition of functions, the identity map is the unit, and an element is invertible if and only if it is a permutation.

By a previous theorem, an element f Q[X] represents an invertible element in the multiplicative monoid of Q[X]/(d) if and only if gcd(f,d) = 1.