On -fibrations of subalgebras of polynomial algebras
Let be a free module over a noetherian ring. For , let be the ideal generated by coefficients of . For an element with , if , there exists such that .This is a generalization of a lemma on the division of forms due to de Rham (Comment. Math. Helv., 28 (1954)) and has some applications to the study of singularities.
The dual of a Gorenstein module is called a co-Gorenstein module, defined by Lingguang Li. In this paper, we prove that if R is a local U-ring and M is an Artinian R-module, then M is a co-Gorenstein R-module if and only if the complex is a minimal flat resolution for M when we choose a suitable triangular subset on R̂. Moreover we characterize the co-Gorenstein modules over a local U-ring and Cohen-Macaulay local U-ring.
In this paper, we use a characterization of -modules such that to characterize Cohen-Macaulay rings in terms of various dimensions. This is done by setting to be the local cohomology functor of with respect to the maximal ideal where is the Krull dimension of .
We prove that for a commutative ring , every noetherian (artinian) -module is quasi-injective if and only if every noetherian (artinian) -module is quasi-projective if and only if the class of noetherian (artinian) -modules is socle-fine if and only if the class of noetherian (artinian) -modules is radical-fine if and only if every maximal ideal of is idempotent.
In this paper we study commutative rings whose prime ideals are direct sums of cyclic modules. In the case is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring , the following statements are equivalent: (1) Every prime ideal of is a direct sum of cyclic -modules; (2) where is an index set and is a principal ideal ring for each ; (3) Every prime ideal of is a direct sum of at most...
If is a domain with the ascending chain condition on (integral) invertible ideals, then the group of its invertible ideals is generated by the set of maximal invertible ideals. In this note we study some properties of and we prove that, if is a free group on , then is a locally factorial Krull domain.
Let be a ring with an identity (not necessarily commutative) and let be a left -module. This paper deals with multiplication and comultiplication left -modules having right -module structures.
We shall prove that if is a finitely generated multiplication module and is a finitely generated ideal of , then there exists a distributive lattice such that with Zariski topology is homeomorphic to to Stone topology. Finally we shall give a characterization of finitely generated multiplication -modules such that is a finitely generated ideal of .