Ring Operations on Coset Space of Ideal/Examples/Integer Multiples of 5

Example of Use of Ring Operations on Coset Space of Ideal
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 $\Z / \ideal 5$.

Proof
$\ideal 5$ is a principal ideal of the ring $\Z$.

In the ring $\Z / \ideal 5$ we have:
 * $\ideal 5 \circ \ideal 5 = \ideal 5$

However, in $\powerset \Z$, we have $\ideal 5 \circ_\PP \ideal 5 = \ideal {25}$.