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March 2nd, 2018, 08:23 AM   #1
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Ring counter example

let R be the set of functions which maps complex numbers to complex numbers. For $f,g\in R$ define functions where

\[(f+_{R}g)(t)=f(t)+g(t)\] and
\[(f X_{R}g)(t)=(f \circ g)(t)=f(g(t))\]

so that $+_{R}$ is the usual pointwise addition of functions but $X_{R}$ is the composition of functions.
Show that R is not a ring with respect to these operations.

what would be the easiest way to prove this? By contradicting the ring distributivity laws? Also how would i go about doing this.
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Last edited by Jaket1; March 2nd, 2018 at 08:28 AM.
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March 2nd, 2018, 10:18 AM   #2
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Originally Posted by Jaket1 View Post
what would be the easiest way to prove this? By contradicting the ring distributivity laws? Also how would i go about doing this.
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Distributivity is the obvious way to go. Look at some simple examples, it's hard NOT to find a counterexample once you do this.
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