Screened in room. Ridge beam recommendation?
Depending on snow load, if any, “dead load” (composition of roof structure), species of wood, grade of beam, etc., I’d use a 4x8...minimum...for a 13’ span.
The Southern Forest Products Association publishes Souther Pine Size Selection Tables that will help with this question. There are different tables for different snow loads etc. The tables spans start at 10' x 16', which requires a minimum built-up beam of two 2x12's for the ridge beam. A 4x4 is quite a bit less supportive and will likely sag.
It is possible that additional support can be had by adding framing to turn the rafters into trusses etc., but that is a more complicated question.