Finitely Generated k-algebra - Nature of the finite generation - basic question

In Dummit and Foote Chapter 15 on page 657 we find the following definition of a k-algebra:

**Let k be a field. A ring R is a k-algebra if k is contained in the centre of R and the identity of k is the identity of R.**

This defintion is followed by the definition of a finitely generated k-algebra:

**The ring R is a finitely generated k-algebra if R is generated as a ring by k together with some finite set of elements of R.**

I wish to make sure that I understand the idea of the finite generation of the ring R.

I am assuming that we have as generators, the set of elements of k, namely (may be infinite!)

and also the ring elements

So the generated elements would be the those elements themselves plus the elements formed by combining these elements using (possibly repeated) the ring operations of addition and multiplication ...

So typical generated elements would include elements like

and so on ...

Does this look right? Can someone please confirm that I am understanding this correctly.

Peter