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Let R be a ring with identity, and let x = R. Then the equation (-1)x= -x is true in R.

Let R be a ring with identity, and let x = R. Then the equation (-1)x= -x is true in R.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let R be a ring with identity, and let x = R. Then the equation (-1)x= -x is true in R. Fill in the white boxes to make a proof of this fact. For each step in the proof, fill in the grey box to indicate which ring axiom or other fact is used to derive that step. Proof: • By • By • By By • By ● the multiplicative identity law the additive identity law we have we have we have we have we have (-1)x= -X. (You should fill all the boxes, so if you came up with a proof that doesn't fill them all, try including more intermediate steps. Please feel free to look up in the lecture notes what "Proposition 5.11" is.) uniqueness of additive inverses Proposition 5.11 › " the distributive law the additive commutative law the additive inverse law (-1+1)x= 0 (-1)x+ 1x=0|| 0((-x) + x) = 0 0x = 0 (-1)(-x) = 1x (-1)x+x=0
Transcribed Image Text:Let R be a ring with identity, and let x = R. Then the equation (-1)x= -x is true in R. Fill in the white boxes to make a proof of this fact. For each step in the proof, fill in the grey box to indicate which ring axiom or other fact is used to derive that step. Proof: • By • By • By By • By ● the multiplicative identity law the additive identity law we have we have we have we have we have (-1)x= -X. (You should fill all the boxes, so if you came up with a proof that doesn't fill them all, try including more intermediate steps. Please feel free to look up in the lecture notes what "Proposition 5.11" is.) uniqueness of additive inverses Proposition 5.11 › " the distributive law the additive commutative law the additive inverse law (-1+1)x= 0 (-1)x+ 1x=0|| 0((-x) + x) = 0 0x = 0 (-1)(-x) = 1x (-1)x+x=0
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