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Let R be a commutative Noetherian ring of dimension d andB=R[X_1,\ldots,X_m,Y_1^{\pm 1},\ldots,Y_n^{\pm 1}] a Laurent polynomial ringover R. If A=B[Y,f^{-1}] for some f\in R[Y], then we prove the followingresults:  (i) If f is a monic polynomial, then Serre dimension of A is \leq d. In casen=0, this result is due to Bhatwadekar, without the condition that f is a monicpolynomial.  (ii) The p-th Euler class group E^p(A) of A, defined by Bhatwadekar and RajaSridharan, is trivial for p\geq max \{d+1, \dim A -p+3\}. In case m=n=0, thisresult is due to Mandal-Parker.

Serre dimension and Euler class group of overrings of polynomial rings