(a)
z|x . So x = az , for some a in A.
z|y . So y = bz , for some b in A.
Therefore x/z + y/z = a + b.
Also x+y = az + bz = (a+b)z
So (x+y)/z = a+b = x/z + y/z.
(b)
z|x . So x = az , for some a in A.
y.(x/z) = y.a
Also (y.x) = y.az
So (y.x)/z = y.az/z = y.a = y.(x/z)
(c)
y|z. So z= cy , for some c in A.
x|(z/y) so c = z/y = kx for some k in A.
z = cy = kx.y = k(x.y) ( since c = kx)
So x.y | z.
Clearly, z/(x.y) = k.
z/y = c = kx
So (z/y)/x = kx/x = k
Therefore z/(x.y) = (z/y)/x
(Here A must be integral domain. Otherwise we can not conclude
those.)
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