Exercise 3.10

What is $(a+b)(a-b)$ in a ring? What is $(ab{)}^3$

Answer

$(a+b)(a-b)=(a+b)a-(a+b)b=a^2+ba-ab-b^2$. It is not necessarily $a^2-b^2$ since the multiplication is not commutative.

$(ab{)}^3=abaabab$. It is not necessarily $a^3{b}^3$ since the multiplication is not commutative.