Question

Define a new multiplication in $\mathit{Z}$by the rule:$\mathit{a}\mathit{b}\mathbf{=}\mathbf{0}$ for all $\mathit{a}\mathit{b}\mathbf{\in }\mathit{Z}$. Show that with ordinary addition and new multiplication, $\mathit{Z}$is a commutative ring.

Short answer

It is proved that $Z$is a ring.

Step-by-step solution

Prove under addition

As we know that $Z$under addition satisfies closure for addition, associative addition, commutative addition, and additive identity. The property which states $a\in Z$that for each , the equation $a+x=0_z$has solution.

Now, prove under multiplication.

Prove under multiplication

Prove that $Z$is closed on multiplication and it follows associative and distributive law:

1. Closed under multiplication:

Consider $a,b\in Z$which implies that $ab=0\in Z$. Therefore, it is closed under multiplication.

2. Associative property:

Consider $a,b,c\in Z$which implies that $a\left(bc\right)=0=ab$. Therefore, it satisfies associative property under multiplication.

3. Distributive Law:

Consider $a,b,c\in Z$which implies that:

$a\left(b+c\right)=0=ab+ac$

and

$\left(a+b\right)c=ac+bc$

Therefore, it satisfies distributive property under distributive property.

Hence, it is proved that $Z$is a ring.