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March 2nd, 2018, 08:23 AM  #1 
Senior Member Joined: Jan 2016 From: Blackpool Posts: 104 Thanks: 2  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. Thanks Last edited by Jaket1; March 2nd, 2018 at 08:28 AM. 
March 2nd, 2018, 10:18 AM  #2 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706  

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