Let R and S be commutative rings, and o: R S a ring homo- morphism....

 

Transcribed Image Text:

Let R and S be commutative rings, and o: R → S a ring homo- morphism. Let : R[X] → S[X] be given by a) Show that b) Show that c) Show that n Caixi i=0 = 2 Σφ(αι)Χ. i=0 is a ring homomorphism. is injective if and only if is injective. is surjective if and only if y is surjective. Problem 3. Let V be a vector space over a field k, let WC V be any subset, and let kW = {a₁w₁ + a₂w₂+...+an Wn | ai e k, w₁ € W}. Show that kW is a subspace of V. Let R and S be commutative rings, and o: R → S a ring homo- morphism. Let : R[X] → S[X] be given by a) Show that b) Show that c) Show that n Caixi i=0 = 2 Σφ(αι)Χ. i=0 is a ring homomorphism. is injective if and only if is injective. is surjective if and only if y is surjective. Problem 3. Let V be a vector space over a field k, let WC V be any subset, and let kW = {a₁w₁ + a₂w₂+...+an Wn | ai e k, w₁ € W}. Show that kW is a subspace of V.

Answer rating: 100% (QA)

a To demonstrate that the mapping 0 0 0 RXSX provided by 0 0 i0 n an i X i i0 n ya i Since X i is a ring homomorphism we must confirm that it respects addition and multiplication and maintains the rin... View the full answer