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Exercise 11.
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Elements Of Modern Algebra
- Let be the ring of Gaussian integers. Let divides and divides. Show that is an idea of. Show that is a maximal ideal of.arrow_forwardLet R be as in Exercise 1, and show that the principal ideal I=(2)={2n+m2|n,m} is a maximal ideal of R. Exercise 1. 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.arrow_forward29. 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
- Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )arrow_forwardAccording 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_forwardIn the ring of integers, prove that every subring is an ideal.arrow_forward
- 32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .arrow_forward12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .arrow_forwardLet I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,