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
g ??
G 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 ?? } at 0, consisting of open subgroups, such that
G/
G ?? 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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