A description of Hilbert $C^\ast $-modules in which all closed submodules are orthogonally closed

Inert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules

Introduction Suppose that  is a commutative ring with identity,  is a unitary -module and  is a multiplicatively closed subset of .  Factorization theory in commutative rings, which has a long history, still gets the attention of many researchers. Although at first, the focus of this theory was factorization properties of elements in integral domains, in the late nineties the theory was gener...

Some Properties of I-Closed Modules

The State-space of the Lattice of Orthogonally Closed Subspaces

The notion of a strongly dense inner product space is introduced and it is shown that, for such an incomplete space S (in particular, for each incomplete hyperplane of a Hilbert space), the system F(S) of all orthogonally closed subspaces of S is not stateless, and the state-space of F(S) is affinely homeomorphic to the face consisting of the free states on the projection lattice corresponding ...

Modules whose hereditary pretorsion classes are closed under products

A module M is called product closed if every hereditary pretorsion class in σ[M ] is closed under products in σ[M ]. Every module which is locally of finite length is product closed and every product closed module is semilocal. LetM ∈ R-Mod be product closed and projective in σ[M ]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invarian...

Closed Terminologies in Description Logics

We introduce a predictive concept recognition methodology for description logics based on a new closed terminology assumption. During knowledge engineering, our system adopts the standard open terminology assumption as it automatically classifies concept descriptions into a taxonomy via subsumption inferences. However, for applications like configuration, the terminology becomes fixed during pr...