Quotient module

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Definition

Let $(R,*,+,0)$ be a ring (with or without unity) and let $M$ a (left) $R$-module. Let $A\subseteq M$ be a submodule of $M$. Then:

• The quotient group $\frac{M}{A}$ is actually a (left) module too with the operations:
  1. (ADDITION) - given by the quotient group part
     • $+:\frac{M}{A}\times\frac{M}{A}\rightarrow\frac{M}{A}$ by $+:([x],[y])\mapsto [x+y]$
  2. Multiplication/module action: $\cdot:R\times\frac{M}{A}\rightarrow\frac{M}{A}$ by $\cdot:(r,[x])\mapsto [rx]$

Furthermore, if $M$ is a unital module then so is $\frac{M}{A}$

With this we get a canonical projection, $\pi:M\rightarrow\frac{M}{A}$ that is a module homomorphism:

• $\pi:x\mapsto [x]$

and the kernel is $A$.

Characteristic property of the quotient module

Characteristic property of the quotient module/Statement

Proof

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Very important for everything

See also

• Quotient group
• Quotient ring
• Quotient