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 $$c$$ 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

 

[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