Is a ridge beam necessary for a small shed roof?
A ridge board isn't a "beam", per se. It's often just a nice way to bring the rafters together. A proper beam would have a supporting post at each end. 

You don't need a ridge beam, but you do need a way to keep the walls from spreading. This can be accomplished with "collar ties", boards spanning the shed at ceiling height, or with large gussets further up the rafter slope. One collar tie would be enough in your case. 

I built my 12x10 shed by using 4' by ~18" OSB gussets at the peak of each 2x4 rafter set, essentially resulting in a truss, which proved very solid:



Source: http://halbergengineering.blogspot.com/2010/10/structural-engineering-for-minnesota.html