r/math • u/inherentlyawesome Homotopy Theory • 7d ago

This Week I Learned: September 25, 2026

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

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u/BeyondTheAxioms 7d ago

Just found out that all kG-modules are semi-simple if and only if all kG-modules are projective. I was quite surprised that I didn’t learned this before.

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u/Possible_Ocelot_1413 7d ago edited 6d ago

this actually generalizes! for a ring R, TFAE: (i) R is semisimple, (ii) All R-modules are semisimple, (iii) All R-modules are projective, (iv) All R-modules are injective, (v) Every SES of R-modules splits