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 $\circ$
From Left Cosets are Equal iff Product with Inverse in Subgroup, we have:
\(\ds x_1 + J\) | \(=\) | \(\ds x_2 + J\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_1 + \paren {-x_2}\) | \(\in\) | \(\ds J\) |
and:
\(\ds y_1 + J\) | \(=\) | \(\ds y_2 + J\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y_1 + \paren {-y_2}\) | \(\in\) | \(\ds J\) |
Hence from the definition of ideal:
\(\ds \paren {x_1 + \paren {-x_2} } \circ y_1\) | \(\in\) | \(\ds J\) | ||||||||||||
\(\ds x_2 \circ \paren {y_1 + \paren {-y_2} }\) | \(\in\) | \(\ds J\) |
Thus:
\(\ds \paren {x_1 + \paren {-x_2} } \circ y_1 + x_2 \circ \paren {y_1 + \paren {-y_2} }\) | \(\in\) | \(\ds J\) | as $\struct {J, +}$ is a group | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_1 \circ y_1 + \paren {-\paren {x_2 \circ y_2} }\) | \(\in\) | \(\ds J\) | Various ring properties | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x_1 \circ y_1 + J\) | \(=\) | \(\ds x_2 \circ y_2 + J\) | 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:
Ring Axiom $\text A$: Addition forms an Abelian 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.
$\Box$
Ring Axiom $\text M0$: Closure under Product
By definition of $\circ$ in $R / J$, it follows that $\struct {R / J, \circ}$ is closed.
$\Box$
Ring Axiom $\text M1$: Associativity of Product
Associativity can be deduced from the fact that $\circ$ is associative on $R$:
\(\ds \forall x, y, z \in R: \, \) | \(\ds \) | \(\) | \(\ds \paren {x + J} \circ \paren {\paren {y + J} \circ \paren {z + J} }\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x + J} \circ \paren {y \circ z + J}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x \circ y \circ z + J\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x \circ y + J} \circ \paren {z + J}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {x + J} \circ \paren {y + J} } \circ \paren {z + J}\) |
$\Box$
Ring Axiom $\text D$: Distributivity of Product over Addition
Distributivity can be deduced from the fact that $\circ$ is distributive on $R$:
\(\ds \forall x, y, z \in R: \, \) | \(\ds \) | \(\) | \(\ds \paren {\paren {x + J} + \paren {y + J} } \circ \paren {z + J}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {x + y + J} \circ \paren {z + J}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {x + y} \circ z} + J\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {x \circ z} + \paren {y \circ z} } + J\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {x \circ z} + J} + \paren {\paren {y \circ z} + J}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\paren {x + J} \circ \paren {z + J} } + \paren {\paren {y + J} \circ \paren {z + J} }\) |
$\Box$
Having verified all of the ring axioms, it follows that $\struct {R / J, +, \circ}$ is a ring.
$\blacksquare$
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.2$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 60.2$ Factor rings: $\text{(i)}$