(3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ........ O integral domain O field not semi simple ring
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- If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
- Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]
- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]