Sum of Cosets of Ideals is Sum in Quotient Ring

Theorem

Let $\struct {R, +, \circ}$ be a ring.

Let $\powerset R$ be the power set of $R$.

Let $J$ be an ideal of $R$.

Let $X$ and $Y$ be cosets of $J$.

Let $X +_\PP Y$ be the sum of $X$ and $Y$, where $+_\PP$ is the operation induced on $\powerset R$ by $+$.

The sum $X +_\PP Y$ in $\powerset R$ is also their sum in the quotient ring $R / J$.

Proof

As $\struct {R, +, \circ}$ is a ring, it follows that $\struct {R, +}$ is an abelian group.

Thus by Subgroup of Abelian Group is Normal, all subgroups of $\struct {R, +, \circ}$ are normal.

So from the definition of quotient group, it follows directly that $X +_\PP Y$ in $\powerset R$ is also the sum in the quotient ring $R / J$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old