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Math Help - finiteness of a non-commutative ring

  1. #1
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    finiteness of a non-commutative ring

    Let R be a non-commutative ring . Suppose that the number of non-units of R is finite . Can we say that R is a finite ring?
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    I think that we can say R is a finite ring. The only way it could be infinite is if the group of units, call it U, were infinite. But then rU for any r\notin U with r\neq 0 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 \mathbb{R}) would destroy the argument.

    In short: yes, unless R is a field!
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    nimon's proof works only for commutative domains because rU is not necessarily infinite even if U is infinite. the answer to xixi's question is also negative for non-commutative rings.

    for example \mathbb{H}, the ring of quaternions over \mathbb{R}, has this property because every non-zero element of \mathbb{H} 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 R be an infinite commutative (resp. non-commutative) ring. suppose that the number of non-unit elements of R is finite. is R necessarily a field (resp. division ring)?
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    Quote Originally Posted by NonCommAlg View Post
    nimon's proof works only for commutative domains because rU is not necessarily infinite even if U is infinite. the answer to xixi's question is also negative for non-commutative rings.

    for example \mathbb{H}, the ring of quaternions over \mathbb{R}, has this property because every non-zero element of \mathbb{H} 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 R be an infinite commutative (resp. non-commutative) ring. suppose that the number of non-unit elements of R is finite. is R necessarily a field (resp. division ring)?
    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 that R is finite?
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