Editing Generalized permutation matrix (section)

== Generalizations ==
One can generalize further by allowing the entries to lie in a [[ring (mathematics)|ring]], rather than in a field. In that case if the non-zero entries are required to be [[unit (ring theory)|units]] in the ring, one again obtains a group. On the other hand, if the non-zero entries are only required to be non-zero, but not necessarily invertible, this set of matrices forms a [[semigroup]] instead.

One may also schematically allow the non-zero entries to lie in a group ''G,'' with the understanding that matrix multiplication will only involve multiplying a single pair of group elements, not "adding" group elements. This is an [[abuse of notation]], since element of matrices being multiplied must allow multiplication and addition, but is suggestive notion for the (formally correct) abstract group <math>G \wr S_n</math> (the wreath product of the group ''G'' by the symmetric group).