Exercise
Let d, a, b,
p, q 
R[X].
Suppose that
a and b are congruent modulo d, and p and q
are congruent modulo d. Then
a and p are congruent modulo d.
a + p and b + q are congruent modulo d.
a2 and b2 are congruent modulo
d2.
ap and bq are congruent modulo d2.
No. Let a = 1, b = 1 + X, p = X, q = -X and d = X in R[X].Then a and p are not congruent modulo d.
This is
true. We have a - b = rd andp -
q = sd. It follows that a + p
-(b + q) = (r + s)d.
No. Let a = 1, b = 1 + X and d = X in R[X]. Then a2 = 1 and b2 = 1 + 2X + X2 are not congruent modulo d2 = X2.
No. Let a = 1, b = 1 + X, p = X, q = -X and d = X in R[X]. Then ap = X and bq = -X -X2 are not congruent modulo d2 = X2.