Exercise 3.2

Define a new ‘multiplication’ operation on $\mathbb{Z}$, by setting $a\odot b=-ab$. Is $(\mathbb{Z},+,\odot )$ a ring according to Definition 3.1? (Prove it is, or explain why it is not).

Answer

Yes, it is a ring. The properties of addition follows from $\mathbb{Z}$. First, we will prove that the multiplication is associative. Let $a,b,c\in \mathbb{Z}$. Then,

$$a\odot (b\odot c)=-a(-bc)=abc=-((-ab)c)=(a\odot b)\odot c$$

Next, we will prove that distributivity holds.

$$\begin{array}{rl}a\odot (b+c)& =-a(b+c)=-(ab+ac)=-ab+(-ac)=a\odot b+a\odot c\\ (a+b)\odot c& =-(a+b)c=-(ac+bc)=-ac+(-bc)=a\odot c+b\odot c\end{array}$$

Lastly, we will prove that the multiplicative identity exists. Choose $-1$ as the identity. Let $a\in \mathbb{Z}$. Then,

$$\begin{array}{r}-1\odot a=-(-1(a))=a\\ a\odot -1=-a(-1)=a\end{array}$$

Thus, $(\mathbb{Z},+,\odot )$ is a ring. $■$