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 $\Z$.

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

$\ideal 5 \circ \ideal 5 = \ideal 5$

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

This needs considerable tedious hard slog to complete it. In particular: Prove the above. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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