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

[Math Amateur]

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:
View attachment 6250In the above text from Bresar, we read the following:

" ... ... Indeed, if \(\displaystyle c\) is a nonzero central element, then \(\displaystyle cA\) must be, as a nonzero idea of \(\displaystyle A\), equal to \(\displaystyle A\). This implies that \(\displaystyle c\) is invertible. ... ... "Can someone please show me exactly why it is the case that \(\displaystyle cA\) being equal to \(\displaystyle A\) implies that \(\displaystyle c\) is invertible ... Help will be appreciated ...

Peter

 

[Euge]

Since $1 \in A$, then $1\in cA$. Thus, there is a $d\in A$ such that $1 = cd$. Since $c$ is central, $cd = dc$. So $cd = 1 = dc$, showing that $c$ is invertible.

 

[Math Amateur]

Since $1 \in A$, then $1\in cA$. Thus, there is a $d\in A$ such that $1 = cd$. Since $c$ is central, $cd = dc$. So $cd = 1 = dc$, showing that $c$ is invertible.

Thanks Euge ... appreciate your help ...

Peter