Get Algebras, rings, and modules : non-commutative algebras and PDF

By Michiel Hazewinkel, Nadiya M. Gubareni

The conception of algebras, jewelry, and modules is among the primary domain names of contemporary arithmetic. normal algebra, extra particularly 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 likelihood skilled within the 20th century. This quantity is a continuation and an in-depth examine, stressing the non-commutative nature of the 1st volumes of Algebras, jewelry and Modules through M. Hazewinkel, N. Gubareni, and V. V. Kirichenko. it's principally self sufficient of the opposite volumes. The correct buildings and effects from previous volumes were awarded during this quantity.

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8. ) A finitely generated torsion-free module M over a Dedekind domain A is projective. A commutative semihereditary integral domain is called a Prüfer domain. 6 we obtain the following theorem which gives other equivalent definition of a Prüfer domain. 9. ) Let A be an integral domain. Then the FSAE: 1. A is a Prüfer domain. 2. Every finitely generated non-zero ideal of A is invertible. 3. Every finitely generated non-zero ideal of A is projective. © 2016 by Taylor & Francis Group, LLC 28 Algebras, Rings and Modules 4.

The following well known theorem characterizes serial rings in terms of finitely presented modules. 1. (Drozd-Warfield Theorem, [67], [320]). g. ) For a ring A the following conditions are equivalent: 1. A is serial; 2. Any finitely presented right A-module is serial; 3. Any finitely presented left A-module is serial. © 2016 by Taylor & Francis Group, LLC 32 Algebras, Rings and Modules Recall that O is a discrete valuation ring7 if it can be embedded into a division ring D with discrete valuation ν such that O = {x ∈ D∗ : ν(x) ≥ 0} ∪ {0}.

9 Semiperfect and Perfect Rings A ring A is called local if it has a unique maximal right ideal. The following theorem gives various equivalent definitions for a ring to be local. ) The following conditions are equivalent for a ring A with Jacobson radical R: 1. 2. 3. 4. 5. A is local; R is the unique maximal right ideal in A; All non-invertible elements of A form a proper ideal; R is the set of all non-invertible elements of A; A/R is a division ring. Note also the following important result about modules over local rings.

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