Quotient Ring is Ring/Quotient Ring Product is Well-Defined
< Quotient Ring is Ring(Redirected from Quotient Ring Product is Well-Defined)
Jump to navigation
Jump to search
Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$.
Let $J$ be an ideal of $R$.
Let $\struct {R / J, +, \circ}$ be the quotient ring of $R$ by $J$.
Then $\circ$ is well-defined on $R / J$, that is:
- $x_1 + J = x_2 + J, y_1 + J = y_2 + J \implies x_1 \circ y_1 + J = x_2 \circ y_2 + J$
Proof
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 |
$\blacksquare$
Also see
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Theorem $41$
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 60.1$ Factor rings