Question

Show that the set$k$ of all constant polynomials in$\mathrm{\mathbb{Z}}\left[x\right]$ is a subring but not an ideal in$\mathrm{\mathbb{Z}}\left[x\right]$ .

Short answer

It is proved$K$ is a subring, but it is not ideal.

Step-by-step solution

Determine Theorem 3.6

Consider that the zero polynomial $z\left(x\right)=0$is in $K$; thus,$K$is non-empty.

Now, suppose $f\left(x\right),g\left(x\right)\in K$. Then $f\left(x\right)=a$and $g\left(x\right)=b$for some $a,b\in \mathrm{\mathbb{Z}}$.

$f\left(x\right)-g\left(x\right)=a-b\in K$

$a-b\in \mathrm{\mathbb{Z}}$is considered a constant polynomial with an integer coefficient.

$f\left(x\right)g\left(x\right)=ab\in K$

$ab\in \mathrm{\mathbb{Z}}$is considered a constant polynomial with an integer coefficient. Thus, both conditions of Theorem 3.6 are satisfied.

Determine whether is ideal or not

It is confirmed that $K$is a subring of $\mathrm{\mathbb{Z}}\left[x\right]$.

Now, consider $i\left(x\right)=x$in $\mathrm{\mathbb{Z}}\left[x\right]$ and$h\left(x\right)=1$ in $K$and note that

$i\left(x\right)h\left(x\right)=x=i\left(x\right)\notin K$

$i\left(x\right)$is not a constant polynomial.

Therefore, for each$r\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]$ and$a\left(x\right)\in K$ . Also,$r\left(x\right)a\left(x\right)\in K$ and $a\left(x\right)r\left(x\right)\in K$, which means$K$ is not ideal.