This study introduces (R,S)⋆-injective modules, highlighting injective envelopes in pure-semisimple rings.
Let f:S→R be a ring extension. We introduce and study the properties of (R,S)⋆-injective modules and the existence of (R,S)⋆-injective envelopes. Moreover, we show that every R-module has an (R,S)-injective envelope when S is a pure-semisimple ring, partly resolving a conjecture proposed by Guo [Comm. Algebra 52 (2024), 2868–2883].
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Xiaolei Zhang (2025) studied this question.
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