Quotient module

In the mathematical field of algebra, a quotient module or factor module is one of the basic constructions of the theory of modules. For a module and a submodule, the quotient module is essentially uniquely determined goal of surjective homomorphism with kernel.

Quotient modules are the analog of the factor space in terms of the theory of vector spaces and factor group in group theory.

Definition

It was a ring. A - ( left ) sub- module and a module, the module is a quotient of the equivalence classes of the set of elements according to the equivalence relation

With clearly defined module structure for which the canonical surjective map is a homomorphism:

Properties

• Isomorphism theorems: For two submodules of a module

• There is a canonical correspondence between isomorphism classes of monomorphisms with the object and isomorphism classes of epimorphisms with source; a monomorphism corresponds to the quotient module, an epimorphism of the module.
• If a module is finitely generated, or it has a finite length, so this is also true for every quotient module.
• If a ( unitary, associative ) algebra, then

• If a ( two-sided ) ideal in, the factor module is the same as the factor ring.

• Module (mathematics)
• Algebra