submodule (original) (raw)

Examples

1. 1. The subsets {0} and T are always submodules of the module T.

There are some operations on submodules. Given the submodules A and B of T, the sum A+B:={a+b∈T:a∈A∧b∈B} and the intersection A∩B are submodules of T.

The notion of sum may be extended for any family {Aj:j∈J} of submodules: the sum ∑j∈JAj of submodules consists of all finite sums ∑jaj where every aj belongs to one Aj of those submodules. The sum of submodules as well as the intersection ⋂j∈JAj are submodules of T. The submodule R⁢X is the intersection of all submodules containing the subset X.

If T is a ring and R is a subring of T, then T is an R-module; then one can consider the product and the quotient of the left R-submodules A and B of T:

• •
  A⁢B:={finite⁢∑νaν⁢bν:aν∈A,bν∈B⁢∀ν}
• •
  [A:B]:={t∈T:tB⊆A}

Also these are left R-submodules of T.