If a, b are elements of a ring and m, n∈Z, show that (na)(mb)=(mn)(ab)

We have to show that if a, b∈R and m,n∈Z, then (na)(mb)=(nm)(ab). Notice that (na)(mb)=(a+....+a)(b+....+b)n×m×=a(b+...+b)+...+a(b+...+b)ms= (ab+...+ab)+...+(ab+...+ab)m×n×=m(ab)+...+m(ab)n×=(nm)(ab) Hence the proof.

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College
Algebra
Abstract algebra

permaneceerc

Answered question

2021-06-18

If a, b are elements of a ring and m, $n \in Z$ , show that $(n a) (m b) = (m n) (a b)$

Answer & Explanation

unett

Skilled2021-06-19Added 119 answers

We have to show that if a, $b \in R$ and $m, n \in Z$ , then $(n a) (m b) = (n m) (a b) .$
Notice that

$(n a) (m b) = (a + . . . . + a) (b + . . . . + b) n \times m \times = a (b + . . . + b) + . . . + a (b + . . . + b) m s =$

$(a b + . . . + a b) + . . . + (a b + . . . + a b) m \times n \times = m (a b) + . . . + m (a b) n \times = (n m) (a b)$

Hence the proof.

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