Question

Question 4: Let ${\left[a\right]}_{\mathbf{n}}$denote the congruence class of the integer a modulon .

(b) Find the kernel of$\mathit{f}$ .

Short answer

The kernel of f is$\left({\left[4\right]}_{12}\right)$

Step-by-step solution

Definition of kernel

Let$f:R\to S$ be a homomorphism of rings. Then, the kernel off is the set

$K=\{r\in R|f\left(r\right)=0_S\}$.

Definition of f

Here,$f:{\mathrm{\mathbb{Z}}}_{12}\to {\mathrm{\mathbb{Z}}}_4$ that sends${\left[a\right]}_{12}$ to ${\left[a\right]}_4$.

Since fis a well-defined surjective homomorphism, so we can directly say that kernel of fis ideal generated by ${\left[4\right]}_{12}$.

Conclusion

So, the Kernel off is$\left({\left[4\right]}_{12}\right)=\{{\left[0\right]}_{12},{\left[4\right]}_{12},{\left[8\right]}_{12}\}.$