Question

Let T be the ring in example 8 and let f, g be given by

$f\left(x\right)=\left\{\begin{array}{l}0ifx\le 2\\ x-2ifx>0\end{array}\right.g\left(x\right)=\left\{\begin{array}{l}2-xifx\le 2\\ 0ifx>0\end{array}\right.$

Show that$f,g\left(x\right)\in T$and that $fg=0_T$. Therefore, T is not integral domain.

Short answer

It is proved that T is not an integral domain

Step-by-step solution

Ring

It is given that T is a ring and two functions f and g defined on T. It can be written as:

$f\left(x\right)=\left\{\begin{array}{l}0ifx\le 2\\ x-2ifx>2\end{array}\right.g\left(x\right)=\left\{\begin{array}{l}2-xifx\le 2\\ 0ifx>2\end{array}\right.$

Integral domain 

If two non-zero elements f and g of ring T gives then, T is not integral domain.

When $x\le 2$then,

$\begin{array}{rcl}f\left(x\right)g\left(x\right)& =& 0\left(2-x\right)\\ & =& 0\end{array}$

When $x>2$then,

$\begin{array}{rcl}f\left(x\right)g\left(x\right)& =& \left(x-2\right)0\\ & =& 0\end{array}$

Since $fg=0$, where $f,g\ne 0$.

Hence, T is not an integral domain.