Let R be a nonzero ring and Endz (R) := {f : R → R | ƒ an additive group homomorphism}. Show that Endz (R) is a ring with addition defined by (ƒ + g) given by (ƒ + g)(x) = f(x) + g(x) for all ¤ ¤ R and multiplication given by composition. Prove that the units in Endz (R) is Autz (R), the group of additive group automorphisms of R.

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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Let R be a nonzero ring and
Endz (R) := {f : R→ R |f an additive group homomorphism}.
Show that Endz (R) is a ring with addition defined by (f + g) given by (ƒ + g)(x) =
f(x) + g(x) for all x E R and multiplication given by composition. Prove that the
units in Endz (R) is Autz (R), the group of additive group automorphisms of R.
Transcribed Image Text:Let R be a nonzero ring and Endz (R) := {f : R→ R |f an additive group homomorphism}. Show that Endz (R) is a ring with addition defined by (f + g) given by (ƒ + g)(x) = f(x) + g(x) for all x E R and multiplication given by composition. Prove that the units in Endz (R) is Autz (R), the group of additive group automorphisms of R.
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