Ideals
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Please answer each part in full:
Let I and J be ideals of R:
(a) Prove that I+J is the smallest ideal of R containing both I and J;
(b) Prove that IJ is an ideal contained in I (intersection) J
(c) Give an example where IJ is not equal to I (intersection) J
(d) Prove that if R is commutative and if I+J=R, then IJ = I (intersection) J.
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Solution Summary
Ideals are investigated for intersections.
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(d) if I + J = R, there are elements x in I, and y in J such that x + y = 1
then for any z in I n J (their intersection) , z = z1 = zx + zy is in IJ since z is in both I and J, i.e. zy
is in IJ, since y is in J, and zx = xz is in IJ since x is in I, and IJ is closed under addition
by (b), the ideal IJ is always contained in their intersection, so this establishes equality
to prove ...
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