# Quotient Ring is Ring

## Theorem

Let $\struct {R, +, \circ}$ be a ring.

Let $J$ be an ideal of $R$.

Let $\struct {R / J, +, \circ}$ be the quotient ring of $R$ by $J$.

Then $R / J$ is also a ring.

## Proof

First, it is to be shown that $+$ and $\circ$ are in fact well-defined operations on $R / J$.

### Well-definition of $+$

From Ideal is Additive Normal Subgroup that $J$ is a normal subgroup of $R$ under $+$.

Thus, the quotient group $\struct {R / J, +}$ is defined, and as a Quotient Group is Group, $+$ is well-defined.

$\Box$

### Well-definition of $\circ$

 $\displaystyle x_1 + J$ $=$ $\displaystyle x_2 + J$ $\displaystyle \leadsto \ \$ $\displaystyle x_1 + \paren {-x_2}$ $\in$ $\displaystyle J$

and:

 $\displaystyle y_1 + J$ $=$ $\displaystyle y_2 + J$ $\displaystyle \leadsto \ \$ $\displaystyle y_1 + \paren {-y_2}$ $\in$ $\displaystyle J$

Hence from the definition of ideal:

 $\displaystyle \paren {x_1 + \paren {-x_2} } \circ y_1$ $\in$ $\displaystyle J$ $\displaystyle x_2 \circ \paren {y_1 + \paren {-y_2} }$ $\in$ $\displaystyle J$

Thus:

 $\displaystyle \paren {x_1 + \paren {-x_2} } \circ y_1 + x_2 \circ \paren {y_1 + \paren {-y_2} }$ $\in$ $\displaystyle J$ as $\struct {J, +}$ is a group $\displaystyle \leadsto \ \$ $\displaystyle x_1 \circ y_1 + \paren {-\paren {x_2 \circ y_2} }$ $\in$ $\displaystyle J$ Various ring properties $\displaystyle \leadsto \ \$ $\displaystyle x_1 \circ y_1 + J$ $=$ $\displaystyle x_2 \circ y_2 + J$ Left Cosets are Equal iff Product with Inverse in Subgroup

$\Box$

Now to prove that $\struct {R / J, +, \circ}$ is a ring, proceed by verifying the ring axioms in turn:

### A: Addition forms a Group

From:

The definition of a quotient group
Quotient Group is Group

it follows that $\struct {R / J, +}$ is a group.

$\Box$

### M0: Closure of Ring Product

By definition of $\circ$ in $R / J$, it follows that $\struct {R / J, \circ}$ is closed.

$\Box$

### M1: Associativity of Ring Product

Associativity can be deduced from the fact that $\circ$ is associative on $R$:

 $\displaystyle \forall x, y, z \in R: \ \$ $\displaystyle$  $\displaystyle \paren {x + J} \circ \paren {\paren {y + J} \circ \paren {z + J} }$ $\displaystyle$ $=$ $\displaystyle \paren {x + J} \circ \paren {y \circ z + J}$ $\displaystyle$ $=$ $\displaystyle x \circ y \circ z + J$ $\displaystyle$ $=$ $\displaystyle \paren {x \circ y + J} \circ \paren {z + J}$ $\displaystyle$ $=$ $\displaystyle \paren {\paren {x + J} \circ \paren {y + J} } \circ \paren {z + J}$

$\Box$

### D: Distributivity of Ring Product over Addition

Distributivity can be deduced from the fact that $\circ$ is distributive on $R$:

 $\displaystyle \forall x, y, z \in R: \ \$ $\displaystyle$  $\displaystyle \paren {\paren {x + J} + \paren {y + J} } \circ \paren {z + J}$ $\displaystyle$ $=$ $\displaystyle \paren {x + y + J} \circ \paren {z + J}$ $\displaystyle$ $=$ $\displaystyle \paren {\paren {x + y} \circ z} + J$ $\displaystyle$ $=$ $\displaystyle \paren {\paren {x \circ z} + \paren {y \circ z} } + J$ $\displaystyle$ $=$ $\displaystyle \paren {\paren {x \circ z} + J} + \paren {\paren {y \circ z} + J}$ $\displaystyle$ $=$ $\displaystyle \paren {\paren {x + J} \circ \paren {z + J} } + \paren {\paren {y + J} \circ \paren {z + J} }$

$\Box$

Having verified all of the ring axioms, it follows that $\struct {R / J, +, \circ}$ is a ring.

$\blacksquare$