Question

Use the First Isomorphism Theorem to show that ${\mathrm{\mathbb{Z}}}_{20}/\left(5\right)\cong {\mathrm{\mathbb{Z}}}_5$.

Short answer

It can be proved that${\mathrm{\mathbb{Z}}}_{20}/\left(5\right)\cong {\mathrm{\mathbb{Z}}}_5$.

Step-by-step solution

First Isomorphism Theorem

Let $f:R\to S$ be a surjective homomorphism of rings with kernel K . Then the quotient ring R/K is isomorphic to S.

Define

Consider a natural projection$\theta :\mathrm{\mathbb{Z}}\to {\mathrm{\mathbb{Z}}}_5$

Such that$\theta \left(k\right)={\left[k\right]}_5$.

Ker $\theta =\left(5\right)$

It contains the ideal$\left(20\right)$.

Induced homomorphism

Consider$\overline{\theta }:{\mathrm{\mathbb{Z}}}_{20}\to {\mathrm{\mathbb{Z}}}_5$

Such that$\overline{\theta }\left({\left[k\right]}_{20}\right)={\left[k\right]}_5$

It is a surjective induced homomorphism.

Kernel of$\overline{\theta }$ is exactly$\left(5\right)$ in${\mathrm{\mathbb{Z}}}_{20}$.

Conclusion

By First theorem of Isomorphism

${\mathrm{\mathbb{Z}}}_{20}/\left(5\right)\cong {\mathrm{\mathbb{Z}}}_5$.