cyclic module

A cyclic module is the simplest kind of module: it is generated by a single element. Pick one element g; if every element of the module is a ring-scalar times g, the module is cyclic and g is a generator. It is the module analog of a line through the origin spanned by one vector, except the scalars now come from a ring.

Every cyclic module looks like a quotient of the ring: M = R . g is isomorphic to R/(I), where I is the annihilator of g — the set of ring elements that kill g. Over a PID that ideal is principal, I = (d), so M is isomorphic to R/(d) for a single element d. The element d is the entire fingerprint of the cyclic module.

In the operator setting R = F[x], a cyclic module is exactly a cyclic subspace Z(v, T): take g = v, and the F[x]-action is the orbit v, Tv, T^2 v, ... The annihilating polynomial d is then the minimal polynomial of T on that subspace, and T restricted to it is a companion matrix C(d). So one cyclic module corresponds to one companion block.

The point: the structure theorem says every finitely generated PID-module is a direct sum of cyclic modules. Cyclic modules are therefore the atoms, and understanding R/(d) completely is understanding the whole classification. Note a cyclic module can be large — being singly generated is about the number of generators, not the dimension.