  This paper is the widely extended version of the publication, appeared in
Proceedings of ISSAC'2009 conference \citep*{ALM09}. We discuss more details on
proofs, present new algorithms and examples. We present a general algorithm for
computing an intersection of a left ideal of an associative algebra over a
field with a subalgebra, generated by a single element. We show applications of
this algorithm in different algebraic situations and describe our
implementation in \textsc{Singular}. Among other, we use this algorithm in
computational $D$-module theory for computing e. g. the Bernstein-Sato
polynomial of a single polynomial with several approaches. We also present a
new method, having no analogues yet, for the computation of the Bernstein-Sato
polynomial of an affine variety. Also, we provide a new proof of the algorithm
by Brian\c{c}on-Maisonobe for the computation of the $s$-parametric annihilator
of a polynomial. Moreover, we present new methods for the latter computation as
well as optimized algorithms for the computation of Bernstein-Sato polynomial
in various settings.
