Problem 6. Let o: C → M₂(R) given by a b ø(a + bi) = ( · (8) a Show that this is an injective ring homomorphism.
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![Problem 6. Let o: CM₂ (R) given by
a
b
(49) :).
a
Show that this is an injective ring homomorphism.
p(a + bi) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0ab11347-5cf5-45ee-b032-ac50fc9e0f95%2Fac079cce-7b01-4cae-9b08-e50d9dce5e86%2F47mdc5b_processed.png&w=3840&q=75)
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- 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?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+y414. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.
- 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 ].Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.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 .)
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