Quotient Ring is Ring

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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 $\circ$

From Left Cosets are Equal iff Product with Inverse in Subgroup, we have:

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

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:

Ideal is Additive Normal Subgroup
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.

$\blacksquare$


Sources