Matrix Rank 1 Semigroup Identities

A finite basis of identities is constructed for the semigroup of all rank 1 n X n matrices over the field. It is worthy to notice that every semigroup of all rank r, r > 1, n x n matrices over a finite field has no finite basis of identities. Let G be an arbitrary variety of groups with a finite basis of identities. A finite basis of identities is constructed for the variety generated by all completely 0-simple semigroups over G-groups.