Show that the zero and the unity are unique in field
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that is axiom unproven truths
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First The textbook I am using (Contemporary Abstract Algebra) already gave a proof for this, but I am trying a different way. I am unsure how to go about one of the cases or if the proof is correct. I would appreciate your help.
Claim: If R is a ring with unity, it is unique. If α∈R is a unity, then its multiplicative inverse is unique.
My proof Let R be a ring with unity. Suppose β1,β2∈R both satisfy the properties of being a unity. Then ∀α∈R we have
1)αβ1=β1α=α
2)αβ2=β2α=α
Hence from (1) and (2)we have
αβ1=αβ2⇒αβ1−αβ2=0⇒α(β1−β2)=0
This means that either α=0 or β1−β2=0
If α=0 (here is my struggle case, anything multiplied by 0 is 0 so I dont know what to say, or how this case implies that β1=β)
If α≠0 then β1−β2=0β which means β1=β2. Hence unity is unique.
The other part im okay with. Thank you
Claim: If R is a ring with unity, it is unique. If α∈R is a unity, then its multiplicative inverse is unique.
My proof Let R be a ring with unity. Suppose β1,β2∈R both satisfy the properties of being a unity. Then ∀α∈R we have
1)αβ1=β1α=α
2)αβ2=β2α=α
Hence from (1) and (2)we have
αβ1=αβ2⇒αβ1−αβ2=0⇒α(β1−β2)=0
This means that either α=0 or β1−β2=0
If α=0 (here is my struggle case, anything multiplied by 0 is 0 so I dont know what to say, or how this case implies that β1=β)
If α≠0 then β1−β2=0β which means β1=β2. Hence unity is unique.
The other part im okay with. Thank you
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