By Michiel Hazewinkel, Nadiya M. Gubareni
The concept of algebras, jewelry, and modules is without doubt one of the primary domain names of recent arithmetic. common algebra, extra in particular non-commutative algebra, is poised for significant advances within the twenty-first century (together with and in interplay with combinatorics), simply as topology, research, and chance skilled within the 20th century. This quantity is a continuation and an in-depth learn, stressing the non-commutative nature of the 1st volumes of Algebras, jewelry and Modules by way of M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. it's principally self reliant of the opposite volumes. The appropriate structures and effects from prior volumes were awarded during this quantity.
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Extra resources for Algebras, Rings and Modules: Non-commutative Algebras and Rings
Kaplansky’s Theorem). g. ) If a ring is a right hereditary then any submodule of a free A-module is isomorphic to a direct sum of right ideals of A. A ring A is said to be right (left) semihereditary if each right (left) finitely generated ideal of A is a projective A-module. If a ring A is both right and left semihereditary, it is called semihereditary. Preliminaries 27 The following theorem gives some of other equivalent condition for a ring to be right (left) semihereditary. 4. ) A ring A is right (left) semihereditary if and only if every finitely generated submodule of a right (left) projective A-module is projective.
Let A be a basic semiperfect ring, and let 1 = e1 + · · · + e s be a decomposition of 1 ∈ A into a sum of mutually orthogonal local idempotents. The simple module Uk = ek A/ek R (resp. Vk = Aek /Rek ) appears in the direct sum decomposition of the module ei R/ei R2 (resp. Rei /R2 ei ) if and only if ei R2 ek (resp. ek R2 ei ) is strictly contained in ei Rek (resp. ek Rei ). , the set of all its submodules is linearly ordered by inclusion. A module is serial if it is decomposed into a direct sum of uniserial submodules.
Any quotient of a right injective A-module is injective. d. ExtkA (X,Y ) = 0 for all k > 1 and all right A-modules X and Y . 2. a. Any semisimple ring is hereditary. b. Any principal ideal domain is hereditary. 3. (Kaplansky’s Theorem). g. ) If a ring is a right hereditary then any submodule of a free A-module is isomorphic to a direct sum of right ideals of A. A ring A is said to be right (left) semihereditary if each right (left) finitely generated ideal of A is a projective A-module. If a ring A is both right and left semihereditary, it is called semihereditary.