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:

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

 

[fresh_42]

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 $c^{-1}$.

 

[Math Amateur]

Thanks ... appreciate your help ...

Peter