Quotient Ring by Null Ideal

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:

$\struct {R, +} / \set {0_R} \cong \struct {R, +}$


This needs considerable tedious hard slog to complete it.
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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 $40$
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