Chapter 6.5, Problem 10E

(a)

To determine

To prove: That the function $\mathbb{Z}\to \mathbb{Z}$ defined by $h\left(x\right)=3x$ is not a ring homomorphism

(b)

To determine

To Prove: That the function $h$ such that ${\mathbb{Z}}_6\to {\mathbb{Z}}_6$ defined by the function $h\left(x\right)=2x$ is not a ring homomorphism.

(c)

To determine

To prove: That the function $g$ such that $\mathbb{Z}\left[x\right]\to \mathbb{Z}$ defined by $g\left(p\left(x\right)\right)=p\left(0\right)$ is a ring homomorphism. Consider $\mathbb{Z}\left[x\right]$ to be a set of polynomials $p\left(x\right)$ in the variable $x$ with integer coefficients.