permaneceerc

## Answered question

2021-06-18

If a, b are elements of a ring and m, $n\in Z$, show that $\left(na\right)\left(mb\right)=\left(mn\right)\left(ab\right)$

### Answer & Explanation

unett

Skilled2021-06-19Added 119 answers

We have to show that if a, $b\in R$ and $m,n\in Z$, then $\left(na\right)\left(mb\right)=\left(nm\right)\left(ab\right).$
Notice that

$\left(na\right)\left(mb\right)=\left(a+....+a\right)\left(b+....+b\right)n×m×=a\left(b+...+b\right)+...+a\left(b+...+b\right)ms=$

$\left(ab+...+ab\right)+...+\left(ab+...+ab\right)m×n×=m\left(ab\right)+...+m\left(ab\right)n×=\left(nm\right)\left(ab\right)$

Hence the proof.

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