Let R be a G-graded commutative ring with identity and let M be a graded R-module. A proper graded submodule N of M is called graded classical prime if for every a, b ∈ h(R), m ∈ h(M ), whenever abm ∈ N , then either am ∈ N or bm ∈ N . The spectrum of graded classical prime submodules of M is denoted by Cl.Specg(M ). We topologize Cl.Specg(M ) with the quasi-Zariski topology, which is analogous to that for Specg(R).2010 MSC: 13A02, 16W50.

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The concept of the M-radical of a submodule B of an R-module A is discussed (R is a commutative ring with identity and A is a unitary fl-module). The M-radical of B is defined as the intersection of all prime submodules of A containing B. The main result of the paper is that if denotes the ideal radical of (B:A), then M-rad B = provided that A is a finitely generated multiplication module. Additionally, it is shown that if A is an arbitrary module, where for some

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We describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on 'schemes', which are terms in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory formation systems.We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth-Bendix completion and recent automatic inductive proof tools.We have evaluated our system in a theory of natural numbers and a theory of lists.

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