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I Simple Unitial Rings .... centre is a field ... ? ...

  1. Dec 2, 2016 #1

    Math Amateur

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    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ...

    I need help with some remarks of Bresar in Example 1.21 on simple unital rings ...

    Example 1.21 reads as follows:


    ?temp_hash=ff70c267b1aa1206a42f97293969b974.png



    In the above text from Bresar, we read the following:

    " ... ... Indeed, if ##c## is a nonzero central element, then ##cA## must be, as a nonzero idea of ##A##, equal to ##A##. This implies that ##c## is invertible. ... ... "


    Can someone please show me exactly why it is the case that ##cA## being equal to ##A## implies that ##c## is invertible ...


    Help will be appreciated ...

    Peter
     

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    Math Amateur, Dec 2, 2016
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  3. Dec 2, 2016 #2

    fresh_42

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    We have ##1 \in A##. Therefore ##0 \neq c = c\cdot 1 \in cA## is a non-trivial ideal of ##A##. Since ##A## is simple, it has to be the entire ring, i.e. ##cA=A##. Now ##1 \in A = cA## means ##1## can be written as ##1=c \cdot a## for some ##a \in A##. We denote this ##a## by ##c^{-1}##.
     
    fresh_42, Dec 2, 2016
  4. Dec 2, 2016 #3

    Math Amateur

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    Thanks ... appreciate your help ...

    Peter
     
    Math Amateur, Dec 2, 2016
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