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:

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



Sources