Some homological characterizations of semigroups and semirings

Principally Quasi-Injective Semimodules

“…A left ℛ-semimodule ℬ is called cyclic if ℬ can be generated by a single element, that is ℬ = 〈 〉 = ℛb = {t b |t ∈ ℛ}for some b∈ ℬ Definition 17 (7). An ℛ-semimodule E is ℬinjective (E is injective relative to ℬ ) if, for each subsemimodule N of ℬ, any ℛ-homomorphism from N to E can be extended to an ℛhomomorphism from ℬ to E. (where i is the inclusion map)…”

“…A left ℛ-semimodule E is injective if it is injective relative to every left ℛ-semimodule. Proposition 18 (7). Let ( ) ∈Ω be an indexed set of a left ℛ-semimodules then ∏ Ω is injective if and only if each is injective for each .…”

“…If ℬ is simple, then every semimodule E is injective relative to ℬ. Remark 21 (7). A semimodule ℬ is quasiinjective if it is ℬ-injective.…”