In post-quantum cryptography, there's a suite of algorithms based on "modular lattice". These schemes are defined in terms of vectors and matrices whose elements are polynomials of a fixed degree (e.g. 256 for ML-KEM/Kyber as well as ML-DSA/Dilithium).
Also, there's the "module" mathematical object, which is a generalization of vector space as I read it.
Then there's a statement in NIST draft for ML-DSA:
When the module $\mathbb{Z}^n_q$ in LWE and SIS is replaced by a module over a ring larger than $\mathbb{Z}_q$ (such as $R_q$), the resulting problems are called MLWE (Module Learning With Errors [14]) and MSIS (Module Short Integer Solution). The security of ML-DSA is based on the MLWE problem over Rq and a nonstandard variant of MSIS called SelfTargetMSIS [15].
which confuses me to the point I submitted a (possibly stupid) comment.
This is mostly a terminology question, so Q: what do these various "modules" mean outside of and relation to lattice-based post-quantum cryptography
As far as I can tell, according to the team behind Kyber and Dilithium, their "module lattice" is a blend between the lattice used in LWE and the lattice used in Ring-LWE. They invented the term and avoided already-existing "modular lattice" which I've mistakenly used towards the beginning of this question.
However, the other modules have established meanings in mathematics. I'm just not sure if this is the correct interpretation.