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 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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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 ...
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