Modules, Submodules

Submodules

A submodule is a module contained in another module. The submodule is a subgroup of the original module, and R carries the submodule into itself.

The intersection of submodules is another submodule. As usual, the submodule generated by a set S is the intersection of submodules containing S. If S contains one element, the submodule is cyclic. If S is finite the submodule is finitely generated.

If a unitary module is generated by S, the module consists of all finite sums of x_is_i, where x_i comes from the ring and s_i is a generator.

Let M be a unitary cyclic module generated by the element s. Let H be the elements of R that drive s into 0. Verify that H is a left ideal, hence the module is isomorphic to the cosets of H. If R is commutative, M = R/H.