Contracting endomorphisms and Gorenstein modules

A finite module M over a noetherian local ring R is said to be Gorenstein if Extⁱ(k, M) = 0 for all i ≠ dim R. An endomorphism φ: R → R of rings is called contracting if for some i ≥ 1. Letting ^φR denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if Hom_R(^φR, M) ≅ M and Extⁱ_R(^φR, M) = 0 for 1 ≤ i ≤ depth R. Received: 7 December 2007     

Reduced multiplication modules

An R-module M is called a multiplication module if for each submodule N of M, N = IM for some ideal I of R. As defined for a commutative ring R, an R-module M is said to be reduced if the intersection of prime submodules of M is zero. The prime spectrum and minimal prime submodules of the reduced module M are studied. Essential submodules of M are characterized via a topological property. It is shown that the Goldie dimension of M is equal to the Souslin number of Spec(M)\mbox{\rm Spec}(M). Also a finitely generated module M is a Baer module if and only if Spec(M)\mbox{\rm Spec}(M) is an extremally disconnected space; if and only if it is a CS-module. It is proved that a prime submodule N is minimal in M if and only if for each x ∈ N, Ann(x) \not í (N:M).\mbox{\rm Ann}(x) \not \subseteq (N:M). When M is finitely generated; it is shown that every prime submodule of M is maximal if and only if M is a von Neumann regular module (VNM); i.e., every principal submodule of M is a summand submodule. Also if M is an injective R-module, then M is a VNM.     

Mod-Retractable Rings

A right module M over a ring R is said to be retractable if Hom _R (M, N) ≠ 0 for each nonzero submodule N of M. We show that M ? _R RG is a retractable RG-module if and only if M _R is retractable for every finite group G. The ring R is (finitely) mod-retractable if every (finitely generated) right R-module is retractable. Some comparisons between max rings, semiartinian rings, perfect rings, noetherian rings, nonsingular rings, and mod-retractable rings are investigated. In particular, we prove ring-theoretical criteria of right mod-retractability for classes of all commutative, left perfect, and right noetherian rings.     

Submodules of Direct Sums of Compressible Modules

Let R be a ring. A right R-module M is called “essentially compressible” if it embeds in each of its essential submodules. Also a module X _R is called “completely essentially compressible” if every submodule of X _R is an essentially compressible R-module. In this aricle, it is shown that a right R-module M embeds in a direct sum of compressible right R-modules if and only if M _R is essentially compressible and every nonzero essentially compressible submodule of M _R contains a compressible submodule. Every essentially compressible R-module is shown to be retractable. Moreover, if either R _R has Krull dimension, or R is Morita equivalent to a right duo ring, then a right R-module embeds in a direct sum of compressible right R-modules if and only if it is completely essentially compressible.     

Independence of the Total Reflexivity Conditions for Modules

We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative Noetherian local ring R and a reflexive R-module M such that Ext_Rⁱ(M,R)=0 for all i>0, but Ext_Rⁱ(M^*,R)≠0 for all i>0. Presented by Juergen Herzog Mathematics Subject Classification (2000) 13D07.     

On（N,U）-Coherence of Modules and Rings

Let U be a flat right R-module and N an infinite cardinal number.A left R-module M is said to be (N,U)-coherent if every finitely generated submodule of every finitely generated M-projective module in σ[M] is (N,U)-finitely presented in σ[M].It is proved under some additional conditions that a left R-module M is (N,U)-coherent if and only if Л^Ni∈I U is M-flat as a right R-module if and only if the (N,U)-coherent dimension of M is equal to zero.We also give some characterizations of left (N,U)-coherent dimension of rings and show that the left N-coherent dimension of a ring R is the supremum of (N,U)-coherent dimensions of R for all flat right R-modules U.     

On the prime radical and Baer’s lower nilradical of modules

Let M be a left R-module. In this paper a generalization of the notion of m-system set of rings to modules is given. Then for a submodule N of M, we define := { m ε M: every m-system containing m meets N}. It is shown that is the intersection of all prime submodules of M containing N. We define rad _R (M) = . This is called Baer-McCoy radical or prime radical of M. It is shown that if M is an Artinian module over a PI-ring (or an FBN-ring) R, then M/rad _R (M) is a Noetherian R-module. Also, if M is a Noetherian module over a PI-ring (or an FBN-ring) R such that every prime submodule of M is virtually maximal, then M/rad _R (M) is an Artinian R-module. This yields if M is an Artinian module over a PI-ring R, then either rad _R (M) = M or rad _R (M) = ∩ _i=1 ⁿ for some maximal ideals of R. Also, Baer’s lower nilradical of M [denoted by Nil_* ( _R M)] is defined to be the set of all strongly nilpotent elements of M. It is shown that, for any projective R-module M, rad _R (M) = Nil_*( _R M) and, for any module M over a left Artinian ring R, rad _R (M) = Nil_*( _R M) = Rad(M) = Jac(R)M. This research was in part supported by a grant from IPM (No. 85130016). Also this work was partially supported by IUT (CEAMA). The author would like to thank the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.     

On Artinian generalized local cohomology modules

Let R be a commutative Noetherian ring with non-zero identity and a be a maximal ideal of R. An R-module M is called minimax if there is a finitely generated submodule N of M such that M/N is Artinian. Over a Gorenstein local ring R of finite Krull dimension, we proved that the Socle of H _a ⁿ (R) is a minimax R-module for each n ≥ 0.     

A note on semiprime right Goldie rings

It is shown that a ring R is semiprime right Goldie if and only if R is right nonsingular and every nonsingular right R-module M has a direct decomposition M = I⊕N, where I is injective and N is a reduced module such that N does not contain any extending submodule of infinite Goldie dimension.     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

On Isolated Submodules

Let R be a ring with identity and let M be a unital left R-module. A proper submodule L of M is radical if L is an intersection of prime submodules of M. Moreover, a submodule L of M is isolated if, for each proper submodule N of L, there exists a prime submodule K of M such that N ? K but L ? K. It is proved that every proper submodule of M is radical (and hence every submodule of M is isolated) if and only if N ∩ IM = IN for every submodule N of M and every (left primitive) ideal I of R. In case, R/P is an Artinian ring for every left primitive ideal P of R it is proved that a finitely generated submodule N of a nonzero left R-module M is isolated if and only if PN = N ∩ PM for every left primitive ideal P of R. If R is a commutative ring, then a finitely generated submodule N of a projective R-module M is isolated if and only if N is a direct summand of M.     

Finiteness properties of minimax and coatomic local cohomology modules

Let R be a noetherian ring, \mathfraka{\mathfrak{a}} an ideal of R, and M an R-module. We prove that for a finite module M, if Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is minimax for all i ≥ r ≥ 1, then Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is artinian for i ≥ r. A local–global principle for minimax local cohomology modules is shown. If Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is coatomic for i ≤ r (M finite) then Hⁱ_\mathfraka(M){{\rm H}^{i}_{\mathfrak{a}}(M)} is finite for i ≤ r. We give conditions for a module which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.     

On weakly prime radical of modules and semi-compatible modules

Summary Let M be a left R-module. Then a proper submodule P of M is called weakly prime submodule if for any ideals A and B of R and any submodule N of M such that ABN ⊆ P, we have AN ⊆ P or BN ⊆ P. We define weakly prime radicals of modules and show that for Ore domains, the study of weakly prime radicals of general modules reduces to that of torsion modules. We determine the weakly prime radical of any module over a commutative domain R with dim (R) ≦ 1. Also, we show that over a commutative domain R with dim (R) ≦ 1, every semiprime submodule of any module is an intersection of weakly prime submodules. Localization of a module over a commutative ring preserves the weakly prime property. An R-module M is called semi-compatible if every weakly prime submodule of M is an intersection of prime submodules. Also, a ring R is called semi-compatible if every R-module is semi-compatible. It is shown that any projective module over a commutative ring is semi-compatible and that a commutative Noetherian ring R is semi-compatible if and only if for every prime ideal B of R, the ring R/\B is a Dedekind domain. Finally, we show that if R is a UFD such that the free R-module R⊕ R is a semi-compatible module, then R is a Bezout domain.     

On Copure Projective Modules and Copure Projective Dimensions

Let R be any ring. A right R-module M is called n-copure projective if Ext¹(M, N) = 0 for any right R-module N with fd(N) ≤ n, and M is said to be strongly copure projective if Ext ⁱ (M, F) = 0 for all flat right R-modules F and all i ≥ 1. In this article, firstly, we present some general properties of n-copure projective modules and strongly copure projective modules. Then we define and investigate copure projective dimensions of modules and rings. Finally, more properties and applications of n-copure projective modules, strongly copure projective modules and copure projective dimensions are given over coherent rings with finite self-FP-injective dimension.     

Relative <Emphasis Type="Italic">FI</Emphasis>-injective and <Emphasis Type="Italic">FI</Emphasis>-flat modules

Let R be a ring, n a fixed nonnegative integer and FP _n (F _n ) the class of all left (right) R-modules of FP-injective (flat) dimensions at most n. A left R-module M (resp., right R-module F) is called n-FI-injective (resp., n-FI-flat) if Ext ¹(N,M) = 0 (resp., Tor ₁(F,N) = 0) for any N ∈ FP _n . It is shown that a left R-module M over any ring R is n-FI-injective if and only if M is a kernel of an FP _n -precover f: A → B with A injective. For a left coherent ring R, it is proven that a finitely presented right R-module M is n-FI-flat if and only if M is a cokernel of an F _n -preenvelope K → F of a right R-module K with F projective if and only if M ∈^⊥ F _n . These classes of modules are used to construct cotorsion theories and to characterize the global dimension of a ring.     

The associated primes of local cohomology modules over rings of small dimension

Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H ⁱ _I (M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H ^{d − 1} _I (M) has finite support for arbitrary R, I, and M. Received: 31 October 2000 / Revised version: 8 January 2001     

Abelian groups as endomorphic modules over their endomorphism ring

Let R be an associative ring with a unit and N be a left R-module. The set M _R(N) = {f: N → N | f(rx) = rf(x), r ∈ R, x ∈ N} is a near-ring with respect to the operations of addition and composition and contains the ring E _R(N) of all endomorphisms of the R-module N. The R-module N is endomorphic if M _R(N) = E _R(N). We call an Abelian group endomorphic if it is an endomorphic module over its endomorphism ring. In this paper, we find endomorphic Abelian groups in the classes of all separable torsion-free groups, torsion groups, almost completely decomposable torsion-free groups, and indecomposable torsion-free groups of rank 2. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 1, pp. 229–233, 2007.     

T-Semisimple Modules and T-Semisimple Rings

We define and investigate t-semisimple modules as a generalization of semisimple modules. A module M is called t-semisimple if every submodule N contains a direct summand K of M such that K is t-essential in N. T-semisimple modules are Morita invariant and they form a strict subclass of t-extending modules. Many equivalent conditions for a module M to be t-semisimple are found. Accordingly, M is t-semisiple, if and only if, M = Z ₂(M) ⊕ S(M) (where Z ₂(M) is the Goldie torsion submodule and S(M) is the sum of nonsingular simple submodules). A ring R is called right t-semisimple if R _R is t-semisimple. Various characterizations of right t-semisimple rings are given. For some types of rings, conditions equivalent to being t-semisimple are found, and this property is investigated in terms of chain conditions.     

Über die Bedingung Going up für {R \subset \hat{R}}

Let (R,\mathfrak m){(R,\mathfrak m)} be a noetherian, local ring with completion [^(R)]{\hat{R}} . We show that R ì [^(R)]{R \subset \hat{R}} satisfies the condition Going up if and only if there exists to every artinian R-module M with Ann_R(M) ì \mathfrakp{{\rm Ann}_R(M) \subset \mathfrak{p}} a submodule U ì M{U \subset M} with Ann_R(U)=\mathfrakp.{{\rm {Ann}}_R(U)=\mathfrak{p}.} This is further equivalent to R being formal catenary, to α(R) = 0 and to H^d_{\mathfrakq/\mathfrakp}(R/\mathfrakp)=0{H^d_{\mathfrak{q}/\mathfrak{p}}(R/\mathfrak{p})=0} for all prime ideals \mathfrakp ì \mathfrakq \subsetneq \mathfrakm{\mathfrak{p} \subset \mathfrak{q} \subsetneq \mathfrak{m}} where d = dim(R/\mathfrakp){d = {\rm {dim}}(R/\mathfrak{p})}.     

Simple-Baer Rings and Minannihilator Modules

Let R be a ring. M is said to be a minannihilator left R-module if r _M l _R (I) = IM for any simple right ideal I of R. A right R-module N is called simple-flat if Nl _R (I) = l _N (I) for any simple right ideal I of R. R is said to be a left simple-Baer (resp., left simple-coherent) ring if the left annihilator of every simple right ideal is a direct summand of _R R (resp., finitely generated). We first obtain some properties of minannihilator and simple-flat modules. Then we characterize simple-coherent rings, simple-Baer rings, and universally mininjective rings using minannihilator and simple-flat modules.