quotient module
A quotient module is what you get by deciding to treat a whole submodule as though it were zero, and then asking what remains. You glue together any two elements that differ by something in the submodule; the leftover identifications form a new, usually smaller, module. It is the module version of forming Z/nZ from Z, or a quotient group, or a quotient ring.
Concretely, let N be a submodule of an R-module M. The cosets m + N partition M, and they inherit operations (m + N) + (m' + N) = (m + m') + N and r(m + N) = rm + N. These are well defined precisely because N is closed under addition and scalar multiplication, and the resulting R-module is written M/N. The map m |-> m + N is a surjective R-module homomorphism M -> M/N with kernel exactly N.
Quotient modules are the right-hand end of every short exact sequence and the engine of the isomorphism theorems: if f : M -> M' is a homomorphism then M/ker(f) is isomorphic to the image of f. Passing to a quotient is the standard way to kill unwanted elements while preserving the module structure — for instance, M/(annihilated part) or M/(torsion) isolates cleaner pieces.
Take M = Z and N = 6Z. The quotient module Z/6Z has six cosets 0+N, 1+N, ..., 5+N, with arithmetic mod 6: this is exactly the cyclic group of order 6 viewed as a Z-module.
Quotienting Z by 6Z produces Z/6Z, the prototypical finite cyclic module.