Letbe a non-commutative ring . Suppose that the number of non-units of
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Let
I think that we can say, were infinite. But then
for any
with
would be an infinite number of non-unital elements, which is a contradiction.
So I guess we must assume that there is at least one non-zero, non-unital element, otherwise an infinite ring of nothing but units with an additive identity thrown in (such as) would destroy the argument.
In short: yes, unless
nimon's proof works only for commutative domains because
for examplethe ring of quaternions over
has this property because every non-zero element of
is a unit. in general, every (infinite) division ring has the property because every
non-zero element of a division ring is a unit. here's a less trivial version of xixi's problem:
let
for my question , I meant there are non-units other than zero and the set of these elements is finite , now by this assumption ,can't we still say thatQuote:
Originally Posted by NonCommAlg
nimon's proof works only for commutative domains because
for example
non-zero element of a division ring is a unit. here's a less trivial version of xixi's problem:
let