Module Theory

1. Front Matter

   Pages i-xiii

   PDF

2. Basic Concepts

   • Alberto Facchini

   Pages 1-32

3. The Krull-Schmidt-Remak-Azumaya Theorem

   • Alberto Facchini

   Pages 33-74

4. Semiperfect Rings

   • Alberto Facchini

   Pages 75-97

5. Semilocal Rings

   • Alberto Facchini

   Pages 99-111

6. Serial Rings

   • Alberto Facchini

   Pages 113-128

7. Quotient Rings

   • Alberto Facchini

   Pages 129-170

8. Krull Dimension and Serial Rings

   • Alberto Facchini

   Pages 171-192

9. Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules

   • Alberto Facchini

   Pages 193-207

10. Biuniform Modules

    • Alberto Facchini

    Pages 209-235

11. Σ-pure-injective Modules and Artinian Modules

    • Alberto Facchini

    Pages 237-266

12. Open Problems

    • Alberto Facchini

    Pages 267-270

13. Back Matter

    Pages 271-288

    PDF

This expository monograph was written for three reasons. Firstly, we wanted to present the solution to a problem posed by Wolfgang Krull in 1932 [Krull 32]. He asked whether what we now call the "Krull-Schmidt Theorem" holds for ar­ tinian modules. The problem remained open for 63 years: its solution, a negative answer to Krull's question, was published only in 1995 (see [Facchini, Herbera, Levy and Vamos]). Secondly, we wanted to present the answer to a question posed by Warfield in 1975 [Warfield 75]. He proved that every finitely pre­ sented module over a serial ring is a direct sum of uniserial modules, and asked if such a decomposition was unique. In other words, Warfield asked whether the "Krull-Schmidt Theorem" holds for serial modules. The solution to this problem, a negative answer again, appeared in [Facchini 96]. Thirdly, the so­ lution to Warfield's problem shows interesting behavior, a rare phenomenon in the history of Krull-Schmidt type theorems. Essentially, the Krull-Schmidt Theorem holds for some classes of modules and not for others. When it does hold, any two indecomposable decompositions are uniquely determined up to a permutation, and when it does not hold for a class of modules, this is proved via an example. For serial modules the Krull-Schmidt Theorem does not hold, but any two indecomposable decompositions are uniquely determined up to two permutations. We wanted to present such a phenomenon to a wider math­ ematical audience.

• modules
• ring theory
• endomorphism ring
• Lattice
• matrices
• Permutation
• ring

"Written in an attractive and fresh mathematical style. Each topic is arranged well and lucidly. The author has made important contributions to the study of direct sum decompositions and many of the main results in this book include his own work."

--EMS Newsletter

• Dipartimento di Matematica e Informatica, Università di Udine, Udine, Italy

  Alberto Facchini

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