By Michiel Hazewinkel, Nadiya M. Gubareni
The conception of algebras, earrings, and modules is without doubt one of the primary domain names of contemporary 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 research, stressing the non-commutative nature of the 1st volumes of Algebras, earrings and Modules through M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. it really is principally self sufficient of the opposite volumes. The suitable structures and effects from past volumes were provided during this quantity.
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Extra resources for Algebras, Rings and Modules: Non-commutative Algebras and Rings
Suppose X,Y are A-modules. The following conditions are equivalent: 1. Y is injective. 2. ExtnA (X,Y ) = 0 for all X and all n > 0. 3. Ext1A (X,Y ) = 0 for all X. 8 Hereditary and Semihereditary Rings A ring A is said to be right (left) hereditary if each right (left) ideal of A is a projective A-module. If a ring A is both right and left hereditary, it is called hereditary. There are many other equivalent definitions of a right (left) hereditary ring. The following theorem gives some of these equivalent conditions.
5. ) A non-zero ideal I of an integral domain A invertible if and only if it is projective. 6. ) Any invertible ideal I of an integral domain A is finitely generated. A commutative hereditary integral domain is called a Dedekind domain. The following theorem gives other equivalent definition of a Dedekind domain. 7. ) Let A be an integral domain. Then the FSAE: 1. A is a Dedekind domain. 2. A is a Noetherian integrally closed domain in which any non-zero prime ideal is maximal. An A-module over a ring A is torsion-free if mx = 0 for m ∈ M, x ∈ A implies m = 0.
A semiperfect semidistributive ring will be written as an SPSD-ring. In the study of the structure of SPSD-rings an important role is played by the following theorems, which give criteria for a ring being right (left) semidistributive. 9. Tuganbaev). ) A semiperfect ring A is right (left) semidistributive if and only if for any local idempotents e, f ∈ A the set eA f is a uniserial right f A f -module (left eAe-module). 10. (Reduction theorem for SPSD-rings). g. ) Let A be a semiperfect ring, and let 1 = e1 + · · · + en be a decomposition of 1 ∈ A into a sum of mutually orthogonal local idempotents.
Algebras, Rings and Modules: Non-commutative Algebras and Rings by Michiel Hazewinkel, Nadiya M. Gubareni