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Ideals and Maximal Ideals

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Please help with the following problems.

a) If I,J are ideals in a ring R such that I+J=R and R is isomorphic to the product ring (R/I)x(R/J) when IJ=0, describe the idempotents corresponding to this product decomposition;
b) Describe the maximal ideal of RxR where in this case R is the set of real numbers;
c) How many roots does x^2-2 have, modulo 8?

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Solution Preview

a) If I,J are ideals in a ring R such that I+J=R and R is isomorphic to the product ring (R/I)x(R/J) when IJ=0, describe the idempotents corresponding to this product decomposition;

Since I+J=R, and the intersection of I and J is 0, we have an isomorphism R--->(R/I)x(R/J). It sends an element a to the pair of its residue classes by I and by J resp. . To find the inverse image of the pair (C,C'), where C and C' are residue classes by I and J resp. , we first have to take any representatives a and b of these classes, then to take ...

Solution Summary

This solution helps with problems involving ideals and maximal ideals. It helps describe idempotents corresponding to product decomposition, describe the maximal ideals and find the number of roots a module has.

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