Every ring without zero divisor it is an integral domain. T OF O *
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- Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here][Type here] True or False Label each of the following statements as either true or false. 3. Every integral domain is a field. [Type here][Type here] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]
- Let D be an integral domain with four elements, D=0,e,a,b, where e is the unity. a. Prove that D has characteristic 2. b. Construct an addition table for D.Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)Prove that every ordered integral domain has characteristic zero.
- 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]Let R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.19. Find a specific example of two elements and in a ring such that and .
- Label each of the following as either true or false. If a set S is not an integral domain, then S is not a field. [Type here][Type here][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]Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]