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Abstract  

Neat subgroups of abelian groups have been generalized to modules in essentially two different ways (corresponding to (a) and (b) in the Introduction); they are in general inequivalent, none implies the other. Here we consider relations between the two versions in the commutative case, and characterize the integral domains in which they coincide: these are the domains whose maximal ideals are invertible.

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Abstract  

The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ∈ Ob Ab | G is a torsion group, and for all gG the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, …,∞}. (o(·) means the order of an element, and n ≤ ∞ means n < ∞.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ∈ Ob CompAb | G has a neighbourhood subbase {G α} at 0, consisting of open subgroups, such that G/G α is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective.

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Fontana, M. and Picozza, G. , On some classes of integral domains defined by Krull’s a.b. operations, J. Algebra , 341 (2011), 179–197. Picozza G

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Abstract

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by , is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J, there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), has an edge joining I and J whenever either I contains a J-regular element or J contains an I-regular element. Here, for every Artinian ring R, we prove that |Max (R)|−1≦ωreg(R))≦|Max (R)| and , where k is the number of fields, appeared in the decomposition of R to local rings. Among other results, we prove that is strongly connected if and only if R is an integral domain. Finally, the diameter and the girth of the regular graph of ideals of Artinian rings are determined.

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), 389 – 396 . [2] Chapman , S. T. , Halter-Koch , F. and Krause , U. , Inside Factorial Monoids and Integral Domains , Journal of

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References [1] Boisen , M. and Sheldon , P. B. , CPI-extensions: overrings of integral domains with special prime spectrums , Canad. J. Math. , 29 ( 1977

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. Amer. Math. Soc. , 125 ( 1997 ), 2853 – 2854 . [5] A nderson , D. D. and Z afrullah , M. , Integral domains in which nonzero locally principal ideals are invertible

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