Semirings of endomorphisms of semigroups form an important class of semirings. All examples which can be found in the literature concern semigroups with a subcommutative operation. We show that there exists a non-subcommutative semigroup whose endomorphisms form a semiring (this answers a question raised by Professor A. H. Clifford). We also give an example of a semigroup whose set of endomorphisms is not embeddable into a semiring; however it is the disjoint union of two semirings, but one of these semirings is not embeddable into a semiring with identity.