scholarly journals Right hereditary affine PI rings are left hereditary

1988 ◽  
Vol 30 (1) ◽  
pp. 115-120
Author(s):  
Ellen Kirkman ◽  
James Kuzmanovich
Keyword(s):  
Dedekind Domain ◽  
Finitely Generated ◽  
Hereditary Ring

Small [11] gave the first example of a right hereditary PI ring which is not left hereditary. Robson and Small [9] proved that a prime PI right hereditary ring is a classical order over a Dedekind domain, and hence is Noetherian (and therefore left hereditary). The authors have shown [4] that a right hereditary semiprime PI ring which is finitely generated over its center is left hereditary. In this paper we consider right hereditary PI rings T which are affine (i.e. finitely generated as an algebra over a central subfield k).

scholarly journals On a generalization of prime submodules of a module over a commutative ring

2017 ◽  
Vol 37 (1) ◽  
pp. 153-168
Author(s):  
Hosein Fazaeli Moghimi ◽  
Batool Zarei Jalal Abadi
Keyword(s):  
Commutative Ring ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Maximal Ideals ◽  
Prime Submodules ◽  
Prime Submodule

‎Let $R$ be a commutative ring with identity‎, ‎and $n\geq 1$ an integer‎. ‎A proper submodule $N$ of an $R$-module $M$ is called‎ ‎an $n$-prime submodule if whenever $a_1 \cdots a_{n+1}m\in N$ for some non-units $a_1‎, ‎\ldots‎ , ‎a_{n+1}\in R$ and $m\in M$‎, ‎then $m\in N$ or there are $n$ of the $a_i$'s whose product is in $(N:M)$‎. ‎In this paper‎, ‎we study $n$-prime submodules as a generalization of prime submodules‎. ‎Among other results‎, ‎it is shown that if $M$ is a finitely generated faithful multiplication module over a Dedekind domain $R$‎, ‎then every $n$-prime submodule of $M$ has the form $m_1\cdots m_t M$ for some maximal ideals $m_1,\ldots,m_t$ of $R$ with $1\leq t\leq n$‎.

Arithmetic modules over generalized Dedekind domains

2020 ◽  
pp. 2250045
Author(s):  
Indah Emilia Wijayanti ◽  
Hidetoshi Marubayashi ◽  
Iwan Ernanto ◽  
Sutopo
Keyword(s):  
Free Module ◽  
Dedekind Domain ◽  
Finitely Generated ◽  
Multiplication Module ◽  
Dedekind Domains ◽  
Torsion Free Module ◽  
Polynomial Module ◽  
Torsion Free

Let [Formula: see text] be a finitely generated torsion-free module over a generalized Dedekind domain [Formula: see text]. It is shown that if [Formula: see text] is a projective [Formula: see text]-module, then it is a generalized Dedekind module and [Formula: see text]-multiplication module. In case [Formula: see text] is Noetherian it is shown that [Formula: see text] is either a generalized Dedekind module or a Krull module. Furthermore, the polynomial module [Formula: see text] is a generalized Dedekind [Formula: see text]-module (a Krull [Formula: see text]-module) if [Formula: see text] is a generalized Dedekind module (a Krull module), respectively.

scholarly journals Projective summands in generators

1982 ◽  
Vol 86 ◽  
pp. 203-209 ◽  
Author(s):  
David Eisenbud ◽  
Wolmer Vasconcelos ◽  
Roger Wiegand
Keyword(s):  
Homomorphic Image ◽  
Polynomial Ring ◽  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Positive Direction ◽  
Chain Conditions ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Category Of Modules

An R-module M is a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M(k) has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free summands locally. Of course, generators can fail to have free summands (e.g., over Dedekind domains), and we ask whether they necessarily have non-zero projective summands. The answer is “yes” for rings of dimension 1, as we point out in § 3, and for the polynomial ring in one variable over a Dedekind domain. In § 1 we show that for 2-dimensional rings the answer is intimately connected with the structure of projective modules. Our main result in the positive direction, Theorem 1.3, grew out of the attempt, in conversations with T. Stafford, to understand the case R = k[x, y]. In § 2 we give examples of rings having generators with no projective summands. The last section contains miscellaneous observations, some of them on rings without chain conditions.

Σ-Rickart modules

2019 ◽  
Vol 19 (11) ◽  
pp. 2050207
Author(s):  
Gangyong Lee ◽  
Mauricio Medina-Bárcenas
Keyword(s):  
Direct Summand ◽  
Endomorphism Ring ◽  
Finitely Generated ◽  
Hereditary Ring ◽  
Direct Sum ◽  
Factor Module ◽  
Rickart Modules

Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50’s. In this paper, we study the notion of a [Formula: see text]-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module [Formula: see text] is called [Formula: see text]-Rickart if every direct sum of copies of [Formula: see text] is Rickart. It is shown that any direct summand and any direct sum of copies of a [Formula: see text]-Rickart module are [Formula: see text]-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring [Formula: see text] is right hereditary if and only if every submodule of any projective right [Formula: see text]-module is projective if and only if every factor module of any injective right [Formula: see text]-module is injective. Also, we have a characterization of a finitely generated [Formula: see text]-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.

scholarly journals Finitely projective modules over a Dedekind domain

1978 ◽  
Vol 26 (3) ◽  
pp. 330-336 ◽  
Author(s):  
V. A. Hiremath
Keyword(s):  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Injective Modules ◽  
Exact Sequences

AbstractAs dual to the notion of “finitely injective modules” introduced and studied by Ramamurth and Rangaswamy (1973), we define a right R-module M to be finitely projective if it is projective. with respect to short exact sequences of right R-modules of the form 0 → A → B → C → 0 with C finitely generated. We have completely characterized finitely projective modules over a Dedekind domain. If R is a Dedekind domain, then an R-module M is finitely projective if and only if its reduced part is torsionless and coseparable.For a Dedekind domain R, finite projectivity, unlike projectivity is not hereditary. But it is proved to be pure hereditary, that is, every pure submodule of a finitely projective R-module is finitely projective.

scholarly journals Half-exact coherent functors over Dedekind domains

2019 ◽  
Vol 18 (05) ◽  
pp. 1950099
Author(s):  
Adson Banda
Keyword(s):  
Prime Ideal ◽  
Principal Ideal ◽  
Dedekind Domain ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Ideal Domain ◽  
Direct Sum ◽  
Dedekind Domains ◽  
Coherent Functor ◽  
Coherent Functors

Let [Formula: see text] be a principal ideal domain (PID) or more generally a Dedekind domain and let [Formula: see text] be a coherent functor from the category of finitely generated [Formula: see text]-modules to itself. We classify the half-exact coherent functors [Formula: see text]. In particular, we show that if [Formula: see text] is a half-exact coherent functor over a Dedekind domain [Formula: see text], then [Formula: see text] is a direct sum of functors of the form [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] is a finitely generated projective [Formula: see text]-module, [Formula: see text] a nonzero prime ideal in [Formula: see text] and [Formula: see text].

Semi-irreducible Zariski spaces of modules

2014 ◽  
Vol 13 (08) ◽  
pp. 1450054
Author(s):  
A. Nikseresht ◽  
A. Azizi
Keyword(s):  
Dedekind Domain ◽  
Finitely Generated ◽  
Dedekind Domains ◽  
The Family ◽  
Finitely Generated Modules ◽  
Torsion Free

In this paper, we state conditions under which, the family of semi-irreducible submodules of a module determine a Zariski space of that module and study the properties of this space. Also we characterize semi-irreducible submodules of finitely generated modules over Dedekind domains. Moreover, assuming that M and M′ are finitely generated modules over a Dedekind domain having isomorphic semi-irreducible Zariski spaces, we find some common properties of M and M′. In particular, we show that in this case the torsion-free components of M and M′ have the same rank and the torsion submodules of [Formula: see text] and [Formula: see text] are isomorphic, where N and N′ are the intersection of all semi-irreducible submodules of M and M′, respectively.

Quasi-strongly Gorenstein projective modules

2019 ◽  
Vol 18 (10) ◽  
pp. 1950182
Author(s):  
Kui Hu ◽  
Fanggui Wang ◽  
Longyu Xu ◽  
Dechuan Zhou
Keyword(s):  
Dedekind Domain ◽  
Projective Modules ◽  
Finitely Generated ◽  
Prüfer Domain ◽  
Dedekind Domains ◽  
Gorenstein Projective Modules ◽  
Semihereditary Rings

In this paper, we introduce the class of quasi-strongly Gorenstein projective modules which is a particular subclass of the class of finitely generated Gorenstein projective modules. We also introduce and characterize quasi-strongly Gorenstein semihereditary rings. We call a quasi-strongly Gorenstein semihereditary domain a quasi-SG-Prüfer domain. A Noetherian quasi-SG-Prüfer domain is called a quasi-strongly Gorenstein Dedekind domain. Let [Formula: see text] be a field and [Formula: see text] be an indeterminate over [Formula: see text]. We prove that every ideal of the ring [Formula: see text] is strongly Gorenstein projective. We also show that every ideal of the ring [Formula: see text] (respectively, [Formula: see text]) is strongly Gorenstein projective. These domains are examples of quasi-strongly Gorenstein Dedekind domains.

Algebras Over Dedekind Domains

1973 ◽  
Vol 25 (4) ◽  
pp. 842-855 ◽  
Author(s):  
Joseph A. Wehlen
Keyword(s):  
Dedekind Domain ◽  
Finitely Generated ◽  
Dedekind Domains ◽  
Inertial Coefficient

The purpose of this paper is two-fold : first, to show that Dedekind domains satisfy a generalization of the Wedderburn-Mal'cev Theorem and, secondly, to classify certain types of finitely generated projective algebras over a Dedekind domain.With respect to the first problem, E. C. Ingraham has shown that a Dedekind domain R is an inertial coefficient ring (IC-ring) if and only if R has zero radical or R is a local Hensel ring.

Baer module hulls of certain modules over a Dedekind domain

2016 ◽  
Vol 15 (08) ◽  
pp. 1650141 ◽  
Author(s):  
Jae Keol Park ◽  
S. Tariq Rizvi
Keyword(s):  
Endomorphism Ring ◽  
Dedekind Domain ◽  
The Other ◽  
Finitely Generated ◽  
Precise Description ◽  
Prime Integer ◽  
Finitely Generated Module ◽  
Baer Ring ◽  
Noetherian Domain ◽  
Infinitely Generated Modules

The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module [Formula: see text], the Baer module hull, [Formula: see text], is the smallest Baer overmodule contained in a fixed injective hull [Formula: see text] of [Formula: see text]. For a certain class of modules [Formula: see text] over a commutative Noetherian domain, we characterize all essential overmodules of [Formula: see text] which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module [Formula: see text] over a Dedekind domain has a Baer module hull if and only if the torsion submodule [Formula: see text] of [Formula: see text] is semisimple. Further, in this case, the Baer module hull of [Formula: see text] is explicitly described. As applications, various properties and examples of Baer hulls are exhibited. It is shown that if [Formula: see text] are two modules with Baer hulls, [Formula: see text] may not have a Baer hull. On the other hand, the Baer module hull of the [Formula: see text]-module [Formula: see text] ([Formula: see text] a prime integer) is precisely given by [Formula: see text]. It is shown that infinitely generated modules over a Dedekind domain may not have Baer module hulls.

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