the collision side of lattice hardness: given a random matrix over a module of polynomial rings, find a short nonzero vector it maps to zero. an attacker who forges a lattice signature or finds a hash collision has solved it — so its hardness prices unforgeability
module-SIS is the signing counterpart of module learning with errors: MLWE hides secrets, MSIS prevents forgeries. together they carry the post-quantum signature schemes of the Dilithium family