In most applications the ridge board is not structural and only serves as a convienience for the carpenters as they frame the roof.
Roof without ridge board.
Because the ridge board is a smaller member typically only an inch thick sometimes even less it is not a structural member and doesn t carry the gravity load.
First off yes it is a common practice to frame a roof with no ridgeboard.
You don t need a ridge beam but you do need a way to keep the walls from spreading.
This video shows how to make roof rafters with a rafter square and or a speed squ.
You may splice two boards together to achieve this length if necessary using a 14 and 1 2 inch block of the same material.
One collar tie would be enough in your case.
A structural ridge is the standard alternative to ceiling joists.
While the joists commonly attach to the side of a ridge beam they can also sit on top of it as shown in the diagram below which creates an interesting look.
Ridge beam sizing is based on the span of the beam between supports and the amount of roof load supported by the beam.
Ridge beams are required by code if the roof slope is less than 3 in 12 irc section r802 3.
This can be accomplished with collar ties boards spanning the shed at ceiling height or with large gussets further up the rafter slope.
A ridge beam carries the loads of the rafters connected to it and must be both strong enough to carry the weight and well supported at the bearing points at each end to transfer the loads down to the ground.
Its only job is to provide a connection point for the roof joists on each side of the roof.
A ridge board is a non structural member that serves as a prop for opposing rafters to rest against and connect to.
The ridge board s length will be the outside measurement of the building plus the length of the overhang at both ends.
A ridge board was not always required and the photo above of the attic in a 1920s era house shows a roof framed without one.
For 2 x 4 rafters use a 2 x 6 ridge board.
Rafters and ridge boards.
In this design the tops of the rafters attach to a load bearing ridge beam that runs along the length of the roof.