Question #351029

3.10 If U, V are ideals of R, let U + V = {u + v | u ∈ U, v ∈ V }.

Prove that U + V is also an ideal.

Expert's answer

Let "u + v \\in U + V" and "r \\in R". Since "U" is an ideal, "\\{ru, ur\\} \\subset R"; since "V" is an ideal, "\\{rv, vr\\} \\subset R". Hence, using left and right distributivity of the ring "R",

"r(u + v) = ru + rv \\in R, \\\\\n(u + v)r = ur + vr \\in R"

(the sums are elements of "R" because the ring is stable under addition). This proves that "U + V" is both a left and right ideal of "R". Therefore, "U + V" is an ideal.

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