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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**Additional resources for Algebras, rings, and modules : non-commutative algebras and rings**

**Sample text**

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.