You could say after the bell rings, or even after the bell or just after bell . It really depends on the terminology used in your school.
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Raen"I will use “bell ring” (is that what it’s called?) where in the beginning of class right after the bell rang, I give students a short quiz consisting of 1~3 questions that review prior lesson."This sentence does not make sense to me. I don't understand the sense in which you are using "where" or how the two parts are connected.