Quotient Ring by Null Ideal
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Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Let $\struct {\set {0_R}, +, \circ}$ be the null ideal of $\struct {R, +, \circ}$.
Let $\struct {R / \set {0_R}, +, \circ}$ be the quotient ring of $R$ defined by $\set {0_R}$.
Then $\struct {R / \set {0_R}, +, \circ}$ is isomorphic to $\struct {R, +, \circ}$.
Proof
Consider the additive group $\struct {R, +}$.
From Trivial Quotient Group is Quotient Group:
- $\struct {R, +} / \set {0_R} \cong \struct {R, +}$
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Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 22$. Quotient Rings: Example $40$