DEFINITION: We define an ideal I C R to be a prime ideal if it is a proper ideal of R with the property that whenever ab e I we have either a e I or b € I. CLAIM: Every maximal ideal J of R is a prime ideal. Use the second claim and the result from homework, Z[x]/(x) = Z, to argue that not every prime ideal is maximal.

Q: Let BCR be a subset of R. Prove that there is an ideal I of R such that B n I has at most one…

A: To show that ideals I and J  exist satisfying conditions 1) and 2)

Q: The ideal I Ð J is a prime ideal of Z Ð Z if O A. I = J O B. I = 0 and J is prime in R2 O C. I = Z…

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Q: Show that the set of nonunits in Z8 is an ideal of Z8

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Q: 2. Let R be as in Exercise 1, and show that the principal ideal I = (V2) = {2n + m √2 \n Є Z, mЄ Z}…

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Q: Problem 4 Let I = (6, 15, 27) be the ideal in Z generated by 6, 15, and 27. Since every ideal in Z…

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Q: Prove that every non-zero prime ideal of Z is a maximal ideal.

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Q: Let S be a prime ideal of R and suppose that a product a₁a₂a3... an, of elements of R, belongs to S.…

A: Let S be a prime ideal of R and suppose that a product , of elements of R, belongs to S. Then prove…

Q: Let A and B be ideals. Prove that A + B = {a+ ß: a E A, ß E B} is an ideal.

A: Let (R, +, ∘) is a ring and A and B are ideals of R.

Q: Suppose D is a PID. Prove that every non-zero prime ideal is maximal.

A: Suppose the R is a PID. We need to show that every non-zero prime ideal is maximal. In case you need…

Q: What is the difference between the notation and {3} .What is the difference between the notation…

A: Here the notation <3> is represent ideal of Z generated by 3      i.e <3> = 3Z is ideal…

Q: Explain why (or prove) {∑nk=3 akxk|ak ∈ R} is not a prime ideal in R[x].

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Q: Every ideal in Q(i√2) [x, y, z] is finitely generated, where Q(i√2) is (isomorphic to) Q[v]/(v² +…

A: The statement is false. Every ideal in the ring ?(i√2)[x, y, z] is not necessarily finitely…

Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

Q: 5Z is a prime ideal of Z. ) 12Z is a maximal ideal of Z

A: An ideal P of a ring R is a called a prime ideal if ab in P implies a or b is in P.

Q: Let S = {a+ bi |a, b e Z , b is even}. Show that S is a subring of Z[i], but not an ideal of Z[i].…

A: Given- S=a+bi| a, b ∈ℤ , b is even .To show that S is a subring of ℤ[i] , but not an ideal of ℤ[i].…

Q: Prove that every ideal in F[x], where F is a field, is a principal ideal

A: To show: Every ideal in F[x], where F is a field is a principle ideal