Abstract
The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:-
-
(a)
To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc.
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(b)
To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules.
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(c)
To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules.
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(d)
To examine the structure of certain classes of abelian groups and to try to find modules with similar structures.
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(e)
To find roles of answers to some of above types of questions in the general theory of modules.
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© 1999 Hindustan Book Agency (India) and Indian National Science Academy
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Singh, S. (1999). Serial Modules and Rings. In: Passi, I.B.S. (eds) Algebra. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9996-3_14
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