  We formulate a substantial improvement on Buchberger's algorithm for
Gr\"obner bases of zero-dimensional ideals. The improvement scales down the
phenomenon of intermediate expression swell as well as the complexity of
Buchberger's algorithm to a significant degree. The idea is to compute a new
type of bases over principal ideal rings instead of over fields like Gr\"obner
bases. The generalizations of Buchberger's algorithm from over fields to over
rings are abundant in the literature. However they are limited to either
computations of strong Gr\"obner bases or modular computations of the numeral
coefficients of ideal bases with no essential improvement on the algorithmic
complexity. In this paper we make pseudo-divisions with multipliers to enhance
computational efficiency. In particular, we develop a new methodology in
determining the authenticity of the factors of the pseudo-eliminant, i.e., we
compare the factors with the multipliers of the pseudo-divisions instead of the
leading coefficients of the basis elements. In order to find out the exact form
of the eliminant, we contrive a modular algorithm of proper divisions over
principal quotient rings with zero divisors. The pseudo-eliminant and proper
eliminants and their corresponding bases constitute a decomposition of the
original ideal. In order to address the ideal membership problem, we elaborate
on various characterizations of the new type of bases. In the complexity
analysis we devise a scenario linking the rampant intermediate coefficient
swell to B\'ezout coefficients, partially unveiling the mystery of hight-level
complexity associated with the computation of Gr\"obner bases. Finally we make
exemplary computations to demonstrate the conspicuous difference between
Gr\"obner bases and the new type of bases.
