Let R be a ring with unity and let End((R, )) be the ring of endomorphisms of (R, ) Let a R, and let a R R be given by a (x) ax for x R a Show that a is an endomorphism of (R, ) b Show that R ' a a R is a subring of End ((R, )) c Prove the analogue of Cayley 's theorem for R by showing that R ' of (b) is isomorphic to R

Question: Let R be a ring with unity and let End((R, +)) be the ring of endomorphisms of (R, +). Let a R, and let

Let R be a ring with unity and let End((R, +)) be the ring of endomorphisms of (R, +). Let a ∈ R, and let λ_a : R → R be given by λ_a(x) = ax for x ∈ R.

a. Show that λ_a is an endomorphism of (R, +).

b. Show that R' = {λ_a | a ∈ R} is a subring of End ((R, +)).

c. Prove the analogue of Cayley's theorem for R by showing that R' of (b) is isomorphic to R.

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a For x y R we have a x y ax y ax ay a x a y Thus a is a homomorphism of R with itself that is ... View full answer

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