On semirings all of whose semimodules have injective envelopes

Abstract

© 2021 World Scientific Publishing Company. It was shown in [Y. Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997) 121-131.] that each semimodule over an additively regular semiring has an injective envelope. We show the converse statement is true as well; moreover, the result holds even if each finitely generated semimodule has an injective envelope.