#In a ring, every element either is inversable or it's nilpotent

80 messages · Page 1 of 1 (latest)

upper stirrup
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My proof for: No element in a ring is simultanously inversable and nilpotent:
Let x be an inversable and nilpotent element. It's pretty obvious that if x is inversable, so is x^n inversable for any n integer (because the inverse is x^(-n)), but there exists a m such that x^m = 0
This implies that 0 is inversable but that's a contradiction.

silent tulip
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it isn't in fields and such

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atleast

upper stirrup
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how is this related to the question

silent tulip
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hence proven :3

rare pulsar
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but not that the non-inversable elements are nilpotent, or that the non-nilpotent elements are inversable

silent tulip
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oh i forgot you gotta prove everytime once in algebra

rare pulsar
upper stirrup
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So any help @rare pulsar

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?

rare pulsar
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i don't know all i knew is that coffey wasn't right, sorry 😔

upper stirrup
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Bruh

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What if I prove that no element is not nipotent nor non-inversable

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Would that be sufficient with my other proof for this?

rare pulsar
zinc sequoia
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in Z, 2 is not invertible or nilpotent

cursive token
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Yes also in the matrice ring not all non invertible matrices are nilpotent

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For example the matrice J with only 1 at every coordinate

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It’s not invertible (every columns is linearly dépend of each other)

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And J^m=n^m * J with n the dimension of the matrice J

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For all m

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So never 0

cursive token
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My bad

upper stirrup
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But wouldn't that be what the question say actually?

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I was able to prove that no element is simultanously invertible and nilpotent

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So that leaves only that elements are either

  1. Neither
  2. Nilpotent but not invertable
  3. Invertable but no nilpotent
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And I got to prove that no elements are neither

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But idk how

blissful sentinel
upper stirrup
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...

blissful sentinel
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Nilpotent and zero divisor are different concepts

upper stirrup
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huh

blissful sentinel
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The question is false, take 2 in Z/Z6

upper stirrup
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I might be little goofy

blissful sentinel
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Yeah

upper stirrup
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I'm getting silly

blissful sentinel
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K

upper stirrup
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So how do I prove this anyway?

blissful sentinel
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Anything productive to say?

upper stirrup
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Please

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omega

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Enlightnen men

blissful sentinel
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Prove what?

upper stirrup
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The question

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Your question

upper stirrup
blissful sentinel
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Prove it yourself

upper stirrup
blissful sentinel
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But anyway yeah, if x is unital, then it can't be a 0 divisor is a fact of all rings

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2 in Z/Z6 isn't a unit since gcd(2,6) != 1 (or do all the multiplications)

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And 2^n is either 2 or 4 in Z/6Z, so it's not nilpotent

upper stirrup
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I will prove that if an element is inversable then it isn't a 0 divisor

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Let x != 0

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Then there exists a y such that xy = yx = 1

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We have to prove that !(Exists y !=0(xy = 0) ) = For every y = 0(xy != 0)

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But this conclusion is false

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Or am I missing something here?

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Think I did the negation statement wrong

blissful sentinel
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Let x be a unit and BWOC suppose its a 0 divisor.
Then there exists non-0 y such that xy=0
Left multiply by x^-1 and you get y=0 contradiction

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So the only thing you can multiply a unit by to get 0 is 0

upper stirrup
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I see

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Is this all you need to finish the proof?

blissful sentinel
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Iirc yes

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Any amount of googling can find a proof

zinc sequoia
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2 is neither a unit nor a zero divisor in Z.

upper stirrup
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Hmm

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Then what is wrong with me proof?

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Wouldn't that have been enough?

zinc sequoia
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What proof?

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The one that someone can't be both a unit and zero divisor?

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Because that's not relevant to the fact that there are things that are neither units nor zero divisors

upper stirrup
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Ok

upper stirrup