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

  • #1
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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:


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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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  • #2
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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}##.
 
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  • #3
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Thanks ... appreciate your help ...

Peter
 

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