Let
Exercise 1.
According to part
is a ring. Assume that the set
is an ideal of
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ELEMENTS OF MODERN ALGEBRA
- Show that the ideal is a maximal ideal of .arrow_forward31. Prove statement of Theorem : for all integers and .arrow_forward34. 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.arrow_forward
- According to part a of Example 3 in Section 5.1, the set R={m+n2|m,n} is a ring. Assume that the set I={a+b2|aE,bE} is an ideal of R, and show that I is not a maximal ideal of R. Example 3 in Section 5.1 The set of all real numbers of the form m+n2 with m and n, is a subring of the ring of all real numbers.arrow_forward23. Find all distinct principal ideals of for the given value of . a. b. c. d. e. f.arrow_forwardLabel each of the following statements as either true or false. The only ideals of the set of real numbers are the ideals of {0} and .arrow_forward
- 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.arrow_forwardLet be as described in the proof of Theorem. Give a specific example of a positive element of .arrow_forwardFind all maximal ideals of .arrow_forward
- 29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .arrow_forward[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage