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.

中文翻译：

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在半环上，所有半模都具有单射包络

它显示在 [Y. Katsov，张量积和半模在加法规则半环上的单射包络，代数科洛克。 4(1997) 121–131.] 加法规则半环上的每个半模都有一个单射包络。我们证明了相反的陈述也是正确的；此外，即使每个有限生成的半模都有一个单射包络，结果仍然成立。