Prove that the ideal <x2 + 1> is prime in Z[x] but not maximal in Z[x].
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A: Given two inequalities 4x2+9y2-24x-90y+225<0 ………………………(1)x2-6x-4y+25≥0…………………………(2)
Prove that the ideal <x2 + 1> is prime in Z[x] but not maximal in Z[x].
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- Show that the ideal is a maximal ideal of .Prove that if R is a field, then R has no nontrivial ideals.34. If is an ideal of prove that the set is an ideal of . The set is called the annihilator of the ideal . Note the difference between and (of Exercise 24), where is the annihilator of an ideal and is the annihilator of an element of.
- Find a principal ideal (z) of such that each of the following products as defined in Exercise 10 is equal to (z). a. (2)(3)(4)(5)(4)(8)(a)(b)18. Find subrings and of such that is not a subring of .Let be the ring of Gaussian integers. Let divides and divides. Show that is an idea of. Show that is a maximal ideal of.
- 31. Prove statement of Theorem : for all integers and .Find all maximal ideals of .Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.