You may (all circles being concentric):
1) Create the two circles, inner and outer, and Object>Compound Path>Make,
2) Create an outer auxiliary circle, extending outside the outermost parts of the arrows to be made,
3) Create an arrowhead at the centre of the circles; to avoid its pointing (too much) outwards, you may place its centre (rather than the hindmost part/base) at the centre,
4) Move a copy of the arrowhead by the desired gap width, cut at the side Anchor Points, and remove the base segment (and the fill) so you have a path consisting of the two angled segments,
5) Move a copy of 4) back on top of the arrowhead 3),
6) Move 3), 4), and 5), into position with the centre between the two circles (by the average of the radii), and rotate all of them together with the auxiliary circle so you get all the arrowheads placed in the right way (120 degrees for 3, 90 degrees for 4, etc),
7) Hide/delete the auxiliary circles,
8) Hide the arrowheads,
9) Select everything (Ctrl/Cmd+A) and Pathfinder>Divide, and then delete the unneded parts, namely the gaps and the funny thingy in the middle,
Now you have the circle parts in a simplified version as they would be without the sides of the arrowhead sticking out (some make it that way),
10) Show the arrowheads and select each with the corresponding circle part and Pathfinder>Unite (or Add to shape area or whatever it is called in your version).
You may colour the arrow/circle parts as desired in connexion with 9) or 10).
Mr. Bugge's instructions are spot on. Here are some visuals to help you and the steps I took (I admit mine are more quick and dirty, his are more precise):
Create a donut shape as Jacob described.
Using the knife tool (under the eraser), divide the circle into the desired number of parts. You can draw a straight line by holding option and shift. Give each part a color.
Create a triangle by using the polygon tool and giving it 3 sides.
Color and arrange each triangle accordingly.
Create a duplicate arrow to be used later.
Use the pathfinder tool to unite the first segment of the circle and the triangle of the same color to create an arrow.
Make the duplicated triangle white and place it behind the arrow you just made and on top of the circle segment of the next color.
Select the white triangle and the circle segment beneath it. Use the pathfinder to "minus front" so the white triangle is subtracted, giving space between the first arrow (red) and the second (purple).
Repeat this process for each arrow.
I would do it in a similar way as Jacob suggested but would create the triangle with the star tool set to 3 sides (or use arrow keys while creating) because it has a point in the middle of the sides that can be used for snapping. I would also use just one circle to use as a snapping target and later expand the width of its stroke. To position the two triangles on the circle I would use the Rotate tool with Smart Guides on as shown in the image. the top row is outline view and the red arrows show the snapping points on the circle with the center of rotation and center of the triangle's base After that rotate the two triangles along the center of the circle by defining it with Alt + click with the Rotate tool on it and rotate with 120° then repeat the transformation again (Ctrl + D). The rest is expanding the stroke, making a compound path with all three white triangles and using Pathfinder's Minus Front on the expanded stroke.