Determine if R is a field an integral domain a unital ring for R=Z/15Z, R=Z/16Z, R=Z/17Z
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- [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]Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]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]
- If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]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] 21. Prove that ifand are integral domains, then the direct sum is not an integral domain. [Type here]
- [Type here] True or False Label each of the following statements as either true or false. 2. Every field is an integral domain. [Type here]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]Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]
- 8. Prove that the characteristic of a field is either 0 or a prime.Assume that the set R={[x0y0]|x,y} is a ring with respect to matrix addition and multiplication. Verify that the mapping :R defined by ([x0y0])=x is an epimorphism from R to Z. Describe ker and exhibit an isomorphism from R/ker toLet be a field. Prove that if is a zero of then is a zero of