##
**Module theory. Endomorphism rings and direct sum decompositions in some classes of modules.**
*(English)*
Zbl 0930.16001

Progress in Mathematics (Boston, Mass.). 167. Basel: Birkhäuser. xii, 285 p. (1998).

The main purpose of this book is to present solutions of two famous open problems in the theory of rings and modules. The author of the book, Alberto Facchini, is an author of these solutions. These problems have been known as Krull’s problem (raised by W. Krull in 1932) and Warfield’s problem (raised by R. B. Warfield in 1975).

Recall that the Krull-Schmidt Theorem says that if \(M_R\) is a right module (over a ring \(R\)) of finite composition length, and \[ M_R=A_1\oplus\cdots\oplus A_n=B_1\oplus\cdots\oplus B_m \] are two decompositions of \(M_R\) as direct sums of indecomposable modules, then \(n=m\) and, after a suitable renumbering of the summands, \(A_i\cong B_i\) for every \(i=1,\dots,n\).

Krull’s problem. Does the Krull-Schmidt Theorem remain true for any Artinian module \(M_R\)?

This problem had been answered affirmatively for several special cases, for example, R. B. Warfield proved in 1969 that the answer to this question is yes if the base ring \(R\) is right Noetherian or commutative.

Since direct sum decompositions of modules correspond to decompositions of their endomorphism rings in a natural way, Krull’s problem is a special case of the problem of determining what kinds of rings can be endomorphism rings of Artinian modules. R. Camps and W. Dicks have proved that the endomorphism ring of an Artinian module is semilocal. This can be used to prove that if \(M\oplus A\cong M\oplus B\) with Artinian module \(M\), then this implies \(A\cong B\). All module-finite algebras over a semilocal Noetherian commutative ring are semilocal, and hence by a result of Camps and Menal it is possible to show that all such module-finite algebras are isomorphic to endomorphism rings of Artinian modules. Using this, Facchini, Herbera, Levy and Vamos were able to construct a counterexample for a negative answer of Krull’s problem (see Section 8.2 in the book).

Warfield’s problem. In 1975, R. B. Warfield published a result which says that every finitely presented module over a serial ring is a direct sum of uniserial modules. In connection with the Krull-Schmidt Theorem, he asked the question: Does the Krull-Schmidt Theorem hold for direct sums of uniserial modules?

In 1996, Facchini has solved this problem completely by giving a counterexample to show that the Krull-Schmidt Theorem fails for serial modules (see Example 9.21 in the book).

Since the Krull-Schmidt Theorem fails for serial modules, the author introduces a weaker form of this theorem which holds for serial modules. Moreover, this “weak Krull-Schmidt Theorem” holds also for direct sums of “biuniform” modules (see Chapter 9 of the book).

The book consists of 11 chapters. The first 10 chapters contain many interesting topics in module and ring theory, such as the Krull-Schmidt-Remark-Azumaya Theorem (an “infinite version” of the Krull-Schmidt Theorem, assuming each direct summand has local endomorphism ring), semiperfect rings, serial rings, quotient rings, Krull dimension, biuniform modules, \(\Sigma\)-pure-injective modules, etc. The last chapter, Chapter 11, contains 21 open problems related to topics discussed in the book.

Besides its research value, this book is a considerable addition to the list of fundamental books in rings and modules. It is written carefully with necessary backgrounds developed in a logical way. There are many fundamental points worked out in the book. All these make the book an excellent contribution to the development of module and ring theory, and a source of reference. Algebraists will certainly enjoy themselves in reading this book.

Recall that the Krull-Schmidt Theorem says that if \(M_R\) is a right module (over a ring \(R\)) of finite composition length, and \[ M_R=A_1\oplus\cdots\oplus A_n=B_1\oplus\cdots\oplus B_m \] are two decompositions of \(M_R\) as direct sums of indecomposable modules, then \(n=m\) and, after a suitable renumbering of the summands, \(A_i\cong B_i\) for every \(i=1,\dots,n\).

Krull’s problem. Does the Krull-Schmidt Theorem remain true for any Artinian module \(M_R\)?

This problem had been answered affirmatively for several special cases, for example, R. B. Warfield proved in 1969 that the answer to this question is yes if the base ring \(R\) is right Noetherian or commutative.

Since direct sum decompositions of modules correspond to decompositions of their endomorphism rings in a natural way, Krull’s problem is a special case of the problem of determining what kinds of rings can be endomorphism rings of Artinian modules. R. Camps and W. Dicks have proved that the endomorphism ring of an Artinian module is semilocal. This can be used to prove that if \(M\oplus A\cong M\oplus B\) with Artinian module \(M\), then this implies \(A\cong B\). All module-finite algebras over a semilocal Noetherian commutative ring are semilocal, and hence by a result of Camps and Menal it is possible to show that all such module-finite algebras are isomorphic to endomorphism rings of Artinian modules. Using this, Facchini, Herbera, Levy and Vamos were able to construct a counterexample for a negative answer of Krull’s problem (see Section 8.2 in the book).

Warfield’s problem. In 1975, R. B. Warfield published a result which says that every finitely presented module over a serial ring is a direct sum of uniserial modules. In connection with the Krull-Schmidt Theorem, he asked the question: Does the Krull-Schmidt Theorem hold for direct sums of uniserial modules?

In 1996, Facchini has solved this problem completely by giving a counterexample to show that the Krull-Schmidt Theorem fails for serial modules (see Example 9.21 in the book).

Since the Krull-Schmidt Theorem fails for serial modules, the author introduces a weaker form of this theorem which holds for serial modules. Moreover, this “weak Krull-Schmidt Theorem” holds also for direct sums of “biuniform” modules (see Chapter 9 of the book).

The book consists of 11 chapters. The first 10 chapters contain many interesting topics in module and ring theory, such as the Krull-Schmidt-Remark-Azumaya Theorem (an “infinite version” of the Krull-Schmidt Theorem, assuming each direct summand has local endomorphism ring), semiperfect rings, serial rings, quotient rings, Krull dimension, biuniform modules, \(\Sigma\)-pure-injective modules, etc. The last chapter, Chapter 11, contains 21 open problems related to topics discussed in the book.

Besides its research value, this book is a considerable addition to the list of fundamental books in rings and modules. It is written carefully with necessary backgrounds developed in a logical way. There are many fundamental points worked out in the book. All these make the book an excellent contribution to the development of module and ring theory, and a source of reference. Algebraists will certainly enjoy themselves in reading this book.

Reviewer: Dinh Van Huynh (Kuweit)

### MSC:

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16S50 | Endomorphism rings; matrix rings |

16P20 | Artinian rings and modules (associative rings and algebras) |

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16P70 | Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras) |