My drum has a very noticeable ring to it when I play at lower dynamics or do rim shots. Any help to try and fix these?

Excessive or dissonant overtones can be a sign that you've got an unevenly tensioned head. Tap at each lug and make sure the pitch at each is the same. If your head's tuned too unevenly, you may want to loosen it, reseat it, and re-tension it while taking care to keep the tension even from lug to lug as you crank little by little.

Also, keep in mind that some ringing can be normal and some drums that sound ringy up close can actually sound pretty good at a distance.

What kind of heads do you have, top & bottom?

Thank you very much for the info.

And in regards to heads, I have an Evans MX5 on bottom, I currently have a Remo White Max,but its in awful condition. But tomorrow I am getting an Evans Hybrid White, and I'm thinking that replacing the old damaged head should help with the overtones. But I could be wrong.

Because you're playing on a well-used head, there's a good chance that that's one of your main culprits to the ringing. As a head gets more and more worn down, the tension tends to become less balanced, especially if it wasn't meticulously tuned.

Also, be sure to check your snares. Guts that are pulled too tightly across the bottom head can resonate with a strong ringing sound. Take those snares and tune them all to a pitch, but make sure that you're not tuning them too tightly. Snares should be tuned and maintained. You wouldn't leave a guitar's strings to sound horrible, would you?

A new batter head ought to help if you make sure to keep it balanced as you torque it. Check the balance on your bottom head, too. Fiber snare side heads like an MX5 or Falam can give a ringy sound, especially if they're really cranked or out of tune.

If your top and bottom heads are in balance, that should help prevent bitter, dissonant overtones. You can also play with the pitch relationship between the top and bottom heads to affect the drum's overall tone.