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Abstract: Two kinds of decompositions of a -module are introduced in this article, a Loewy-decomposition and a primary decomposition. Algorithms are described which allow to construct decompositions for finite-dimensional -modules. Both decompositions rely on the concept of the -module of relative syzygies introduced in the paper. Furthermore an algorithm is described which tests whether two given -modules are isomorphic under the relevant type of transformations. [Copyright &y& Elsevier]

In this paper we study plane conic geometry, particularly different representations of geometric constructions and relations in plane conic geometry, with Cayley and bracket algebras. We propose three powerful simplification techniques for bracket computation involving conic points, and an algorithm for rational Cayley factorization in conic geometry. The factorization algorithm is not a general one, but works for all the examples tried so far. We establish a series of elimination rules for various geometric constructions based on the idea of bracket-oriented representation and elimination, and an algorithm for optimal representation of the conclusion in theorem proving. These techniques can be used in any applications involving brackets and conics. In theorem proving, our algorithm based on these techniques can produce extremely short proofs for difficult theorems in conic geometry. [Copyright &y& Elsevier]