You have a very late-model 1903 Modified rifle - the receiver was probably manufactured in late-December, 1942, so you have a matching barrel. Take a closer look on the stock - I'll bet the "RIA" is really a FJA, which is standard on rifles of your era. It stands for Frank J. Atwood, the head of the ordnance district that contained Remington and Smith Corona. If there is an RIA, it probably would have RIA/EB or /FK and is a rebuild stamp.
The 1903 you have was a transition between the near-original 1903 clones Remington manufactured earlier and the 1903A3s, which were just starting up when your rifle was made.
One more thing -- look on the rear receiver bridge and see if there is any tooling marks for a 1903A3 sight (the sight looks like a M1 carbine sight - the sight won't be there, but the marks for the base may be).
Feel free to ask follow-up questions or comments or email me.
A forum community dedicated to hunters and enthusiasts. Come join the discussion about optics, trails, clubs, gunsmithing, styles, reviews, accessories, classifieds, and more. The friendliest gun forum on the Internet!