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Maximal ideals

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With respect to the ideal

I=<2,x> in Z[x]

I believe this ideal is maximal because one theorem I have read suggests to me that all maximal ideals of Z[x] are in the form <p, f(x)> where p is prime and f(x) is an element of Z[x] and irreducible mod p. It appears that <2,x> fits this description.

Did I understand correctly that <2,x> is maximal, and if so, how would you show this?

Words of explanation appreciated in the proof.

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https://brainmass.com/math/basic-algebra/maximal-ideals-247559

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Actually, a quotient ring of the form A/I is always a commutative ring, if A is commutative. So we have a commutative ring with 2 elements: two equivalence ...

Solution Summary

The solution does a great job of explaining the questions being asked. It draws out the complete multiplication table and explains that a quotient ring of the form A/I is always commutative. The solution is very well explained and it is easy for anyone to follow along. The student should refer to the attachment which preserves all the formatting. The theorem is provied in the attachment. Overall, an excellent response to the question being asked.

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