a b s t r a c tIn this work, the cardinality of the minimal R-covers of finite rings with respect to the RT-metric is established. By generalizing the result in Nakaoka and dos Santos (2010) [1], the minimal cardinalities of 0-short coverings of finite chain rings are calculated. The connection between R-short coverings of rings with respect to the RT-metric and the 0-short coverings of rings is demonstrated, and with the help of this connection, the problem of finding the minimal cardinalities of R-short coverings of finite chain rings is solved.
In this work, we give a characterization of generalizations of prime and primary fuzzy ideals by introducing 2-absorbing fuzzy ideals and 2-absorbing primary fuzzy ideals and establish relations between 2-absorbing (primary) fuzzy ideals and 2-absorbing (primary) ideals. Furthermore, we give some fundamental results concerning these notions.
Let R be a commutative ring with 1 ≠ 0. A proper ideal I of R is a semiprimary ideal of R if whenever a, b ϵ R and ab ϵ I, we have [Formula: see text] or [Formula: see text]; and I is a weakly semiprimary ideal of R if whenever a, b ϵ R and 0 ≠ ab ϵ I, we have [Formula: see text] or [Formula: see text]. In this paper, we introduce a new class of ideals that is closely related to the class of (weakly) semiprimary ideals. Let I(R) be the set of all ideals of R and let [Formula: see text] be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ(L) and δ(J) ⊆ δ(I). Let δ be an expansion function of ideals of R. Then a proper ideal I of R is called a δ-semiprimary (weakly δ-semiprimary) ideal of R if ab ϵ I (0 ≠ ab ϵ I) implies a ϵ δ(I) or b ϵ δ(I). A number of results concerning weakly δ-semiprimary ideals and examples of weakly δ-semiprimary ideals are given.
Noncommutative derivative operators acting on the quantum 3D space in the sense of Manin are introduced. Furthermore, the quantum 3D space is extended by the series expansion of the logarithm of the grouplike generator in the quantum 3D space. We give its differential calculus and the corresponding Weyl algebra. We also obtain algebra of Cartan-Maurer forms on this extension and the corresponding Lie algebra of vector fields. All noncommutative results are found to reduce to those of the standard commutative algebra when the deformation parameter of the quantum 3D space is set to one.
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