91Ó°ÊÓ

Prove In any ring, \(a(b-c)=a b-a c\) and \((b-c) a \pm b a-c a\).

In each of the following, a set \(A\) with operations of addition and multiplication is given. Prove that \(A\) satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary \(a\). \(A\) is the set \(\mathbb{Q}\) of the rational numbers, and the operations are \(\oplus\) and \(\odot\) defined as follows: $$ a \oplus b=a+b+1 \quad a \odot b=a b+a+b $$

Prove In any integral domain, if \(a^{2}=b^{2}\), then \(a=\pm b\).