Let R be a commutative ring and let I and J be ideals in R. Define the sets I + Jand IJ as follows:andI + J = {i + j | i Є I, jЄ J}IJ := {i1j1 + i2j2 + + İkİk | i1, i2, . . ., ik Є I, Ì¹, Ì2, . . ., Ìk Є J}....(a) Prove that In J, I + J, and IJ are ideals of R.(b) If R = F[X], I=(f(X)), and J = (g(X)), find explicitly In J, I + J, and IJ,i.e., for each of them find a generator.

As you asked multiple different type of questions, so according to the rule I have to solve the…

Answered: Let R be a commutative ring and let I…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 24E: 24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set...

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1) Let R and S be rings. Show that the cartesian product of sets R x S = {(r, ) |r e R, s €s} with the addition (r, s) + (r', s') := (r +r', s + s') and the multiplication (r, s) · (r', s') := (r - r', s - s') is a ring.

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