Problem 7. Consider Let : M₂(Z) → Z given by *(())- Show that this is not a ring homomorphism. = = a.
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- 11. Show that defined by is not a homomorphism.Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?Prove statement d of Theorem 3.9: If G is abelian, (xy)n=xnyn for all integers n.
- Let be as described in the proof of Theorem. Give a specific example of a positive element of .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+y4If 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]
- a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].14. Let be an ideal in a ring with unity . Prove that if then .24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)