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

hexacordoK 2021-10-09 Answered
If a, b are elements of a ring and m, n ∈ Z, show that (na) (mb) = (mn) (ab)

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Expert Answer

SabadisO
Answered 2021-10-10 Author has 16742 answers

We have to show that if a,\(\displaystyle{b}∈{R}\) and \(\displaystyle{m},{n}∈{Z}\), then \((na)(mb)=(nm)(ab)\).
Notice that
\((na)(mb)={(a+....+a)(b+....+b)}n \times m\)

\(=a{(b+...+b)}+...+a{(b+...+b)}m\)

\(={{(ab+...+ab)}+...+{(ab+...+ab)}}m \times n\)

\(={m(ab)+...+m(ab)}n\)

\(=(nm)(ab)\)
Hence the proof.

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