Ring Operations on Coset Space of Ideal

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 $+$.

Similarly, let $X \circ_\PP Y$ be the product of $X$ and $Y$, where $\circ_\PP$ is the operation induced on $\powerset R$ by $\circ$.

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

The product $X \circ_\PP Y$ in $\powerset R$ is a subset of their product in $R / J$.

Let $\ideal 5$ denote the set of all integer multiples of $5$.

Then their product $\ideal 5 \circ_\PP \ideal 5$ in $\powerset \Z$ is a proper subset of their product in $\Z / \ideal 5$.

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