What does the following subring of the complex numbers look like

[micromass]

I'm not sure how to describe polynomials of this form

polynomials of what form?

 

[Metric_Space]

the things you described like i/(x)*(x+i)

 

[micromass]

What about it? It's a quotient of the constant polynomial i and the polynomial x(x+i)...
Thus a(x)=i and b(x)=x(x+i)... I don't really see your confusion here.

 

[Metric_Space]

I was trying to figure out a way of writing things not in the subring, other than the way already written in the question

 

[micromass]

Well, take some elements in (x). And put them in the denumerator of the fraction. Then you obtain things not in the subring.

You obtain things in the subring, by taking elements in (x) and putting them in the denumerator of the fraction...