If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all rR

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Q: Let f(x)= x³ + 2x+ 1 and g(x)= 4x + 1 be two polynomials in a ring (Zs[x],+ ,.). Find q(x) and r(x).

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Q: やthe ring KLx,り,z3 wherc K is a field. Prove (x2 - (y1)) that is a K[x, り,そI . Prime ideal of

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Q: Prove that the rings Z3[x]/ and Z3[x]/ are isomorphic

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Q: Theorem 8.3.14 (First Isomorphism Theorem) Let f be a homomorphism of a ring R into a ring R'. Then…

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Q: Hom worke: Consider the ring (Z [√3], +,.), Let A={ a+b√3:a, be Z₂} Is A subring of Z[√3]?

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Q: Explain why the polynomial rings R[x] and C[x] are not isomorphic.

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Q: Exercise: Prove that the homomorphism f:R R of the ring R onto the ring R' is isomorphism if and…

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Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.

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Q: Theorem(3):- ( Third Fundamental Theorem ) If (K, +,.) and (L,+,.) are two ideals in a ring (R, +,.)…

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Q: QUESTION 2 Suppose f:R+ F is a surjective ring homomorphism from a ring R to a field F. Either prove…

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Q: The ring Z is isomorphic to the ring 3Z True False

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Q: Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a…

A: Let R be a ring with unity e, verify the mapping θ:Z→R defined by θx=x.e is a homomorphism If R and…