Why is right ascension measured on a 24 hour scale rather than a 23 hours and 56 minutes scale? From several texts I have read, I have learned that right ascension is measured on a scale of 24 hours. I understand that Earth rotates almost exactly 360 degrees relative to distant objects, which is what RA is used for. The 24 hour system should only apply to the 361 degree rotation period (solar day). Why then do we split the sidereal day of the earth into 24 hours, when if it were split into 23 hours and 56 minutes we could know where an object would be x hours away after x hours?

More (possibly) relevant information:
I understand the Hour Angle HR is the measure of how far Aries is from the local meridian (RA of current meridian of longitude). This way you could calculate how many hours away the object is, but once again, this would not be exactly correct as 4 minutes has to be subtracted from the 24 hour scale, multiplying the whole time to something like 99.7% (14.44 degrees per hour), correct?

Thanks, this is really confusing me and I would appreciate some help. I did see a similar question but that had nothing to do with the sidereal day v. solar day issue I'm having.
Right ascension is a historical oddity.

To specify a point in the sky you need a coordinate system, the one which we have come to use has it's origin at the Point of Aries on the Equator and the Ecliptic (a reasonable choice), It uses the Equator for one axis, and the meridian through the point of Aries for the other, again these are convenient and reasonable.

The coordinate system also requires a unit to be chosen for measuring angular distance. For measuring declension, we use degrees. The more mathematically pure may prefer radians, but degrees are a commonly used unit for measuring angles.

For measuring the angular distance around the equator we have come to use "hours", by dividing the full turn into 24 parts. This is a historical oddity. It has a few conveniences: The sidereal day is only slightly shorter than 24 hours, so a star with RA of 1hr will be due south about 1 hour after a star with a RA of 0hr (not exactly but close enough for a rule of thumb). A telescope with an equatorial mount that is set to make a full turn in 24 hours will track stars well enough for a human observer. 

Dividing the equator into (23+56/60) parts would be inconvenient since its not a whole number. For these reasons Flamsteed used Hours for his catalogue in 1712, and we have followed his tradition.

So the use of "hours" may be roughly related to the apparent motion of the stars, but it is really just a way of specifying an angle, and using 24 is easier than (23+56/60)