Consider the ring $\mathbb{Z}_{15}$. (a)Let $I=\langle 5\rangle$, List the elements of the coset $4+I$ of $I$ in $Z_{15}?$

Question

Consider the ring $\mathbb{Z}_{15}$.

(a) Let $I=\langle 5\rangle$. List the elements of the coset $4+I$ of $I$ in $\mathbb{Z}_{15}.$

(b) Let $I=\langle 5\rangle.$ In the quotient ring $\mathbb{Z}_{15}/I$, $(3+I)(7+I)=\ ?$

My answer:

(b) $$(3+I)(7+I)=21+I=1+I$$

Thank you.

Answer 1

Since $I$ is an ideal generated by $5$, so $$I=\{5r \, | \, r \in \mathbb{Z}_{15}\}=\{0,5,10\}$$ Therefore, the coset $$4+I=\{4+I, 9+I,14+I\}.$$

Your answer for (b) is fine.