While the opponent is shooting a ball, is it legal to push the Basketball through under the ring before it completely enters the Ring, to prevent a Shoot?
No, it is not legal
What you seem to be describing is a violation known as Goaltending. Touching the ball when it is in or over the ring is a violation and the opposing team is awarded the points associated with the shot.
The relevant ruling states:
Section I—A Player Shall Not:
i. Touch the rim, net or ball while the ball is in the net, preventing it from clearing the basket.
PENALTY: If the violation is at the opponent’s basket, the offended team is awarded two points, if the attempt is from the two point zone and three points if it is from the three-point zone. The crediting of the score and subsequent procedure is the same as if the awarded score has resulted from the ball having gone through the basket, except that the official shall hand the ball to a player of the team entitled to the throw-in. If the violation is at a team’s own basket, no points can be scored and the ball is awarded to the offended team at the free throw line extended on either sideline. If there is a violation by both teams, no points can be scored, play shall be resumed by a jump ball between any two opponents in the game at the center circle.