Quotient module

In the mathematical branch of algebra , a quotient module or factor module is one of the basic constructions of the theory of modules . For a module and a sub-module , the quotient module is the essentially unambiguous goal of a surjective homomorphism with a core . ${\displaystyle M}$ ${\displaystyle N\subseteq M}$ ${\displaystyle M/N}$ ${\displaystyle M\to M/N}$ ${\displaystyle N}$

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

definition

Let it be a ring . For a - (left) module and a sub-module , the quotient module is the set of equivalence classes of elements from according to the equivalence relation ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle N\subseteq M}$ ${\displaystyle M/N}$ ${\displaystyle M}$

${\displaystyle m_{1}\equiv m_{2}\mod N\iff m_{1}-m_{2}\in N}$

with the uniquely determined module structure for which the canonical surjective mapping is a homomorphism: ${\displaystyle M\to M/N}$

${\displaystyle a\cdot (m+N)=am+N.}$

properties

• Isomorphism theorems : The following applies to two sub-modules of a module ${\displaystyle M,N}$ ${\displaystyle Q}$

${\displaystyle M/(M\cap N)\cong (M+N)/N.}$

The following applies to sub-modules ${\displaystyle N\subseteq Q\subseteq P}$

${\displaystyle (P/N)/(Q/N)\cong P/Q.}$

• There is a canonical correspondence between isomorphism classes of monomorphisms with target and isomorphism classes of epimorphisms with source ; a monomorphism corresponds to the quotient module , an epimorphism to the sub-module . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle i\colon N\to M}$ ${\displaystyle M/i(N)}$ ${\displaystyle p\colon M\to Q}$ ${\displaystyle \ker p}$
• If a module is finitely generated , or if it has a finite length , this also applies to every quotient module.
• If a (unitary, associative) algebra, then ${\displaystyle B}$ ${\displaystyle A}$

${\displaystyle B\otimes _{A}(M/N)\cong (B\otimes _{A}M)/U;}$

where stands for the image of in . ${\displaystyle U}$ ${\displaystyle B\otimes _{A}N}$ ${\displaystyle B\otimes _{A}M}$

• If a (two-sided) ideal is in , the factor module is the same as the factor ring . ${\displaystyle I}$ ${\displaystyle A}$ ${\displaystyle A/I}$ ${\displaystyle A/I}$