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mikky05v
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So this is a review problem in our book I came across and i really want to understand it but I am just not having any luck, I did some research and found a guide on solving it but that's not really helping either. We didn't talk about unity elements in class and there aren't any examples in our book just this problem. Would anyone be willing to take the time to fully explain this to me and show me how to prove it?
Question:
You may assume that
E=even integers is a Commutative Ring.
Prove that E does NOT have a unity element
Information I found online: It suffices to show that there is at least a single element, n, of E , for which no element of E acts as a unity element for that specific n.
So specifically, consider the case n = 2 and let m = any element of
E. Then m MUST have exactly 1 of 3 possible properties:
so let CASE 1 cover the 1st possible property of m and show that m cannot be a “unity” for 2
and let CASE 2 cover the 2nd possible property of m and show that m cannot be a “unity” for 2
and let CASE 3 cover the 3rd possible property of m and show that m cannot be a “unity” for 2
Conclude that if there is no unity element ‘e’ for 2 such that 2*e=e*2=2
then, of course, there is no unity element for all of
E …
E does NOT have a unity element
Question:
You may assume that
E=even integers is a Commutative Ring.
Prove that E does NOT have a unity element
Information I found online: It suffices to show that there is at least a single element, n, of E , for which no element of E acts as a unity element for that specific n.
So specifically, consider the case n = 2 and let m = any element of
E. Then m MUST have exactly 1 of 3 possible properties:
so let CASE 1 cover the 1st possible property of m and show that m cannot be a “unity” for 2
and let CASE 2 cover the 2nd possible property of m and show that m cannot be a “unity” for 2
and let CASE 3 cover the 3rd possible property of m and show that m cannot be a “unity” for 2
Conclude that if there is no unity element ‘e’ for 2 such that 2*e=e*2=2
then, of course, there is no unity element for all of
E …
E does NOT have a unity element