The W-characteristic set of a polynomial ideal is the minimal triangular set contained in the reduced lexicographical Groebner basis of the ideal. A pair (G,C) of polynomial sets is a strong regular characteristic pair if G is a reduced lexicographical Groebner basis, C is the W-characteristic set of the ideal , the saturated ideal sat(C) of C is equal to , and C is regular. In this talk, we show that for any polynomial ideal I with given generators one can either detect that I is unit, or construct a strong regular characteristic pair (G,C) by computing Groebner bases such that I \subseteq sat(C)= and sat(C) divides I, so the ideal I can be split into the saturated ideal sat(C) and the quotient ideal I:sat(C). Based on this strategy of splitting by means of quotient and ideal computations, a simple algorithm is presented to decompose an arbitrary polynomial set F into finitely many strong regular characteristic pairs, from which two representations for the zeros of F are obtained: one in terms of strong regular Groebner bases and the other in terms of regular triangular sets.