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Semidistributive Modules and Rings

Semidistributive Modules and Rings - Mathematics and Its Applications

Softcover reprint of the original 1st Edition 1998

Paperback (15 Oct 2012)

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Publisher's Synopsis

A module M is called distributive if the lattice Lat(M) of all its submodules is distributive, i.e., Fn(G + H) = FnG + FnH for all submodules F,G, and H of the module M. A module M is called uniserial if all its submodules are comparable with respect to inclusion, i.e., the lattice Lat(M) is a chain. Any direct sum of distributive (resp. uniserial) modules is called a semidistributive (resp. serial) module. The class of distributive (resp. semidistributive) modules properly cont.ains the class ofall uniserial (resp. serial) modules. In particular, all simple (resp. semisimple) modules are distributive (resp. semidistributive). All strongly regular rings (for example, all factor rings of direct products of division rings and all commutative regular rings) are distributive; all valuation rings in division rings and all commutative Dedekind rings (e.g., rings of integral algebraic numbers or commutative principal ideal rings) are distributive. A module is called a Bezout module or a locally cyclic module ifevery finitely generated submodule is cyclic. If all maximal right ideals of a ring A are ideals (e.g., if A is commutative), then all Bezout A-modules are distributive.

Book information

ISBN: 9789401061360
Publisher: Springer Netherlands
Imprint: Springer
Pub date:
Edition: Softcover reprint of the original 1st Edition 1998
Language: English
Number of pages: 357
Weight: 593g
Height: 240mm
Width: 160mm
Spine width: 20mm