Commutative Rings Whose Finitely Generated Ideals Decompose into Direct Sums of Cyclic Ideals

The question of which commutative rings have the property that every finitely generated module is a direct sum of cyclic modules has been around for many years. We will call these rings FGC-rings. The problem originated in I. Kaplanskys papers  [5], [6];in which it was shown that a local domain is FGC if and only if it is an almost maximal valuation ring. For several years, this is one of the major open problems in the theory. The problem studied and solved in 1970s by various authors (R. S. Pierce[7], Brandal [2], Shores-R. Wiegand [8], S. Wiegand [10], Brandal-R. Wiegand [3] and V´amos [9], leading to a complete solution of the problem in the commutative case. To show that a commutative FGC-ring cannot have an infinite number of minimal prime ideals required the study of topological properties (so-called Zariski and patch topologies). For complete and more leisurely treatment of this subject, see Brandal [1]. It gives a clear and detailed exposition for the reader wanting to study the subject. The main result reads as follows:

Theorem. A commutative ring R is an FGC-ring exactly if it is a finite direct sum of commutative rings of the following kinds:

  1. maximal valuation rings;
  2. almost maximal B´ezout domains;
  3. so-called torch rings (see [2] or [8] for more details on the torch rings).

Therefore, an interesting natural question of this sort is: “What is the class of commutative rings R for which every finitely generated  ideal is a direct sum of principal ideals?” This problem is still open. The ring  R is said to be an almost FGC-ring if every finitely generated ideal decomposes into a direct sum of cyclic (principal)ideals. The purpose of this proposal is to describe all the almost FGC-rings; i.e., characterize the almost FGC-rings and give as many examples as possible.

References

[1] W. Brandal, Commutative Rings Whose Finitely Generated Modules decompose, Lecture Notes in Mathematics, Vol. 723 (Springer, 1979).

[2] W. Brandal, Almost maximal integral domains and finitely generated modules. Trans. Amer. Math. Soc. 183 (1973), 203-222.

[3] W. Brandal and R. Wiegand, Reduced rings whose finitely generated modules decompose. Comm. Algebra 6(2) (1978), 195-201

[4] A. Facchini, On the structure of torch rings. Rocky Mountain J. Math. 13(3) (1983), 423-428.

[5] I. Kaplansky, Elementary divisors and modules. Trans. Amer. Math. Soc. 66,(1949). 464-491.

[6] I. Kaplansky, Modules over Dedekind rings and valuation rings. Trans. Amer.Math. Soc. 72 (1952), 327-340.

[7] R. S. Pierce, Modules over commutative regular rings. Mem. Amer. Math. Soc. 70 (1967).

[8] T. S. Shores and R. Wiegand, Rings whose finitely generated modules are direct sums of cyclics. J. Algebra 32 (1974), 152-172.

[9] P. V´amos, The decomposition of finitely generated modules and fractionally self-injective rings. J. London Math. Soc. 16 (1977), 209-220.

[10] R. Wiegand and S. Wiegand, Commutative rings whose finitely generated modules are direct sums of cyclics. Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976), pp. 406423. Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977.

Related Papers:

1- M. Behboodi (Joint with G. Behboodi Eskandari) On rings over which every finitely generated module is a direct sum of cyclic modules, Hacettepe J. Mathematics and Statistics (Accepted) (ISI)

Abstract: In this paper we study (non-commutative) rings R over which every nitely generated left module is a direct sum of cyclic modules (called left FGC-rings). The commutative case was a well-known problem studied and solved in 1970s by various authors. It is shown that a Noetherian local left FGC-ring is either an Artinian principal left ideal ring, or an Artinian principal right ideal ring, or a prime ring over which every two-sided ideal is principal as a left and a right ideal. In particular, it is shown that a Noetherian local duo-ring R is a left FGC-ring if and only if R is a right FGC-ring, if and only if, R is a principal ideal ring.

2- M. Behboodi (Joint with G. Behboodi Eskandari) Local duo-rings whose finitely generated modules are direct sums of cyclics, Indian J. Pure and Appl. Math. (Accepted) (ISI)

In this paper, we give an answer to the following question of I. Kaplansky[14] in the local case: For which duo rings R is it true that every nitely generated left R-module can be decomposed as a direct sum of cyclic modules? More precisely, we prove that for a local duo ring R, the following are equivalent:  (i) Every nitely generated left R-module is a direct sum of cyclic modules; (ii) Every 2-generated left R-module is a direct sum of cyclic modules;  (iii) Every factor module of RR  R is a direct sum of cyclic modules; (iv) Every factor module of RR  R is serial; (v) Every nitely generated left R-module is serial; (vi) R is uniserial and for every non-zero ideal I of R, R=I is a linearly compact left R-module; (vii) R is uniserial and every indecomposable injective left R-module is left uniserial; and, (viii) Every nitely generated right R-module is a direct sum of cyclic modules.

 

 

 

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