In the article 1, Page 46, the author notes that:
At this point it should be noted that there is an obvious connection with commutative rings. In fact, if $R$ is a commutative domain and $T= C$ is the field of fractions of $R$, the results obtained in Section $2$ concerning prime ideals are well known.
For a commutative ring $R$, what are the prime (maximal) ideals of the polynomial ring $R[x]$? Can they be characterized by their generators?
Even if you know the prime ideals of a commutative ring $R$ very well, there is nothing substantial you can say about the prime ideals of $R[x]$: if there were algebraic geometers would be out of business!
For example it is very easy to determine the prime ideals of $R=\mathbb C[y,z]$ but essentially impossible to classify the height-$2$ prime ideals of $R[x]$, which correspond to curves in $\mathbb A^3_\mathbb C$
Even calculating the minimal number of generators (which can be arbitrarily large, as shown by Macaulay) for such an ideal is a daunting task.
As to the article you refer to: the author is interested only in the very special case where $R$ is a "prime ring", a concept he does not define and which probably means a ring without strict subring. This an extremely restricted class of rings, to say the least.