Expert answers to "If a, b are elements of a ring..."

College
Algebra
Abstract algebra

hexacordoK

Answered question

2021-10-09

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

Answer & Explanation

SabadisO

Skilled2021-10-10Added 108 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
$(n a) (m b) = (a + . . . . + a) (b + . . . . + b) n \times m$

$= a (b + . . . + b) + . . . + a (b + . . . + b) m$

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

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

$= (n m) (a b)$
Hence the proof.

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