Displaying similar documents to “Primary decomposition of torsion R [ X ] -modules.”

Torsion Part of ℤ-module

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

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In this article, we formalize in Mizar [7] the definition of “torsion part” of ℤ-module and its properties. We show ℤ-module generated by the field of rational numbers as an example of torsion-free non free ℤ-modules. We also formalize the rank-nullity theorem over finite-rank free ℤ-modules (previously formalized in [1]). ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [23] and cryptographic systems with lattices [24].

Extending modules relative to a torsion theory

Semra Doğruöz (2008)

Czechoslovak Mathematical Journal

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An R -module M is said to be an extending module if every closed submodule of M is a direct summand. In this paper we introduce and investigate the concept of a type 2 τ -extending module, where τ is a hereditary torsion theory on Mod - R . An R -module M is called type 2 τ -extending if every type 2 τ -closed submodule of M is a direct summand of M . If τ I is the torsion theory on Mod - R corresponding to an idempotent ideal I of R and M is a type 2 τ I -extending R -module, then the question of whether...

On torsionfree classes which are not precover classes

Ladislav Bican (2008)

Czechoslovak Mathematical Journal

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In the class of all exact torsion theories the torsionfree classes are cover (precover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory...