Question

Question 9: $\mathit{R}\mathbf{=}\left\{\left(\begin{array}{cc}a& o\\ b& c\end{array}\right)|a,b,c\in \mathbb{Z}\right\}$ is a ring with identity by Example 19 in Section 3.1.

(b) What is the kernel of f.

Short answer

Answer:

Kernel of is$fis\{\left(\begin{array}{cc}0& 0\\ \mathrm{b}& \mathrm{c}\end{array}\right):\mathrm{b},\mathrm{c}\in \mathrm{\mathbb{Z}}\}$

Step-by-step solution

Find kernel of f

Consider if$f:R\to S$given by $f\left(\begin{array}{cc}a& 0\\ b& c\end{array}\right)=a$

Then, Ker is such that $f\left(\begin{array}{cc}a& 0\\ b& c\end{array}\right)=0$

This implies that a=0.

Conclusion

Kernel of is$fis\{\left(\begin{array}{cc}0& 0\\ \mathrm{b}& \mathrm{c}\end{array}\right):\mathrm{b},\mathrm{c}\in \mathrm{\mathbb{Z}}\}$