Answer to Question #351028 in Abstract Algebra for Fed

Question

3.9 Show that the commutative ring D is an integral domain if

and only if for a, b, c ∈ D with a #= 0 the relation ab = ac implies that

b = c.

Answer 1

ab= ac$\implies$ ab-ac= 0

$\implies$ a(b-c)=0

$\implies$ a=0 or b-c=0

$\implies$ b-c=0 a≠0 $\implies$ b=c