The creature will deal its normal bite damage due to the ring of rat fangs plus the creature's full Strength bonus when the creature uses only its natural attacks. The Universal Monster Rules on natural attacks says
Primary attacks are made using the creature’s full base attack bonus and add the creature’s full Strength bonus on damage rolls.
However, this changes if the wearer of the ring of rat fangs also makes attacks with weapons. The Universal Monster Rules continue, saying that
Creatures with natural attacks and attacks made with weapons can use both as part of a full attack action (although often a creature must forgo one natural attack for each weapon clutched in that limb, be it a claw, tentacle, or slam). Such creatures attack with their weapons normally but treat all of their available natural attacks as secondary attacks during that attack, regardless of the attack’s original type.
So, just biting and clawing and tentacles and whatever other natural attacks you possess? Full Strength bonus. But biting and swinging a sword? Normal sword attacks, but the bite attack suffers a −5 penalty on the attack roll and deals its 1d3 or 1d4 points of damage plus only ½ the creature's Strength modifier.