Artinian cofinite modules over complete Noetherian local rings

Let (R, m) be a complete Noetherian local ring, I an ideal of R and M a nonzero Artinian R-module. In this paper it is shown that if p is a prime ideal of R such that dim R/p = 1 and (0:M p) is not finitely generated and for each i ? 2 the R-module Ext _R ⁱ (M,R/p) is of finite length, then the R-module Ext _R ¹ (M, R/p) is not of finite length. Using this result, it is shown that for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (N,M) are of finite length, if and only if, for all finitely generated R-modules N with Supp(N) ? V (I) and for all integers i ? 0, the R-modules Ext _R ⁱ (M,N) are of finite length.     

Self-injective Right Artinian Rings and Igusa Todorov Functions

We show that a right artinian ring R is right self-injective if and only if ψ(M)?=?0 (or equivalently ?(M)?=?0) for all finitely generated right R-modules M, where ψ, $\phi :\!\!\!\! \mod R \to \mathbb N$ are functions defined by Igusa and Todorov. In particular, an artin algebra Λ is self-injective if and only if ?(M)?=?0 for all finitely generated right Λ-modules M.     

Gorenstein flatness and injectivity over Gorenstein rings

Let R be a Gorenstein ring.We prove that if I is an ideal of R such that R/I is a semi-simple ring,then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical.In addition,we prove that if R→S is a homomorphism of rings and SE is an injective cogenerator for the category of left S-modules,then the Gorenstein flat dimension of S as a right R-module and the Gorenstein injective dimension of E as a left R-module are identical.We also give some applications of these results.     

On the normal ideals of exchange rings

An ideal I of a ring R is called normal if all idempotent elements in I lie in the center of R. We prove that if I is a normal ideal of an exchange ring R then: (1) R and R/I have the same stable range; (2) V(I) is an order-ideal of the monoid C(Specc(R), N), where Specc(R) consists of all prime ideals P such that R/P is local.     

On the depth of certain graded rings associated to an ideal

Let (R,m) be a local ring and Ian ideal of R. In this work we find conditions on Ithat allow us to describe simple relations among depth R(It), depth gr_I(R), depth S(I) and depth S(I/I ²). These relations are useful also from a practical point, of view since it is usually difficult to evaluate depth gr_I(R) and depth S(I/I ²) even with the help of a computer. Furthermore we study the class of ideals that satisfy one of the required conditions and we show that ideals generated by quadratic sequences are in this class     

Cofiniteness and finiteness of local cohomology modules over regular local rings

Let (R,m) be a commutative Noetherian regular local ring of dimension d and I be a proper ideal of R such that mAss _R (R/I) = Assh _R (_I). It is shown that the R- module H^ht(I) _I (R) is I-cofinite if and only if cd(I,R) = ht(I). Also we present a sufficient condition under which this condition the R-module H ⁱ _I (R) is finitely generated if and only if it vanishes.     

Some results on the local cohomology of minimax modules

Let R be a commutative Noetherian ring with identity and I an ideal of R. It is shown that, if M is a non-zero minimax R-module such that dim Supp H _I ⁱ (M) ? 1 for all i, then the R-module H _I ⁱ (M) is I-cominimax for all i. In fact, H _I ⁱ (M) is I-cofinite for all i ? 1. Also, we prove that for a weakly Laskerian R-module M, if R is local and t is a non-negative integer such that dim Supp H _I ⁱ (M) ? 2 for all i < t, then Ext _R ^j (R/I,H _I ⁱ (M)) and Hom _R (R/I,H _I ^t (M)) are weakly Laskerian for all i < t and all j ? 0. As a consequence, the set of associated primes of H _I ⁱ (M) is finite for all i ? 0, whenever dim R/I ? 2 and M is weakly Laskerian.     

<Emphasis Type="Bold">K</Emphasis><Subscript>0</Subscript> of Semiartinian Von Neumann Regular Rings. Direct Finiteness Versus Unit-Regularity

If R is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) R is unit-regular, (2) every factor ring of R is directly finite, (3) the abelian group K _O(R) is free and admits a basis which is in a canonical one to one correspondence with a set of representatives of simple right R-modules. For the class of semiartinian and unit-regular rings the canonical partial order of K _O(R) is investigated. Starting from any partially ordered set I, a special dimension group G(I) is built and a large class of semiartinian and unit-regular rings is shown to have the corresponding K _O(R) order isomorphic to G(P r i m _R ), where P r i m _R is the primitive spectrum of R. Conversely, if I is an artinian partially ordered set having a finite cofinal subset, it is proved that the dimension group G(I) is realizable as K _O(R) for a suitable semiartinian and unit-regular ring R.     

<Emphasis Type="Italic">V</Emphasis>-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and _S V _R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom _R (I _V (R),?) and Hom _R (?,I _V (R)) exact exact complex \( \cdots \to {I_1}\xrightarrow{{{d_0}}}{I_0} \to {I^0}\xrightarrow{{{d_0}}}{I^1} \to \cdots \) of V-injective modules I _i and I ⁱ , i ∈ N₀, such that N ? Im(I ₀ → I ⁰). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A _V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I _V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N ⊕ E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if \(Ext_{{I_V}\left( R \right)}^{ \geqslant n + 1}\left( {I,N} \right) = 0\) for all modules I with finite I _V (R)-injective dimension.     

The unique rank property among gabriel localizations

It is not uncommon for rings to have Gabriel localizations which do not possess the unique rank (UR) property although the rings themselves do have UR. We show that if F is a Gabriel filter of right ideals on a ring R and R_F is the corresponding Gabriel localization, then free R_F?modules of ranks m and n are isomorphic if and only if some F-dense submodule of (R/T_f(R))^m is isomorphic to some F-dense submodule of (R/T_F(R))ⁿ, where T_F(R) is the F-torsion ideal of R.     

General symbolic powers: A homological point of view

Let I be an ideal of a Noetherian ring R and let S be a multiplicatively closed subset of R. We define the n-th (S)-symbolic power of 7 as S(Iⁿ) = IⁿR_s ∩R. The purpose of this paper is to compare the topologies defined by the adic {I_n}_n≤0 and the (S)-symbolic filtration {S(Iⁿ)}n≥o using the direct system {Extⁱ _R(R/Iⁿ,R)}_n≥0     

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R)≤n (n≥2) if and only if the gl right Proj-dim MR≤n - 2 if and only if Extn-1(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R)≤n (n≥1) if and only if every (n-1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R)≤2 are characterized.     

Weighted sums of squares in local rings and their completions, I

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext¹_R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?_R ?_{i ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{i ? I}{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.     

Wakamatsu tilting modules with finite injective dimension

Let R be a left Noetherian ring, S a right Noetherian ring and _R ω a Wakamatsu tilting module with S = End( _R ω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ? 0, we prove that l.id _R (ω) = r.id _S (ω) ? n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.     

The cleanness of (symbolic) powers of Stanley-Reisner ideals

Let Δ be a pure simplicial complex on the vertex set [n] = {1,..., n} and I _Δ its Stanley-Reisner ideal in the polynomial ring S = K[x ₁,..., x _n]. We show that Δ is a matroid (complete intersection) if and only if S/I _Δ ^(m) (S/I _Δ ^(m)) is clean for all m ∈ N and this is equivalent to saying that S/I _Δ ^(m) (S/I _Δ ^(m), respectively) is Cohen-Macaulay for all m ∈ N. By this result, we show that there exists a monomial ideal I with (pretty) cleanness property while S/I ^m or S/I ^m is not (pretty) clean for all integer m ≥ 3. If dim(Δ) = 1, we also prove that S/I _Δ ⁽²⁾ Δ (S/I _Δ ²) is clean if and only if S/I _Δ ⁽²⁾ (S/I _Δ ², respectively) is Cohen-Macaulay.     

Relative copure injective and copure flat modules

Let R be a ring, n a fixed nonnegative integer and I_n (F_n) the class of all left (right) R-modules of injective (flat) dimension at most n. A left R-module M (resp., right R-module F) is called n-copure injective (resp., n-copure flat) if (resp., ) for any N∈I_n. It is shown that a left R-module M over any ring R is n-copure injective if and only if M is a kernel of an I_n-precover f:A→B of a left R-module B with A injective. For a left coherent ring R, it is proven that every right R-module has an F_n-preenvelope, and a finitely presented right R-module M is n-copure flat if and only if M is a cokernel of an F_n-preenvelope K→F of a right R-module K with F flat. These classes of modules are also used to construct cotorsion theories and to characterize the global dimension of a ring under suitable conditions.     

TOPOLOGICAL CHARACTERIZATIONS OF THE EXTENDING PROPERTY OF RINGS

A commutative ring R is called extending if every ideal is essential in a direct summand of RR. The following results are proved： （1） C（X） is an extending ring if and only if X is extremely disconnected; （2） Spec（R） is extremely disconnected and R is semiprime if and only if R is a nonsingular extending ring; （3） Spec（R） is extremely disconnected if and only if R/N（R） is an extending ring, where N（R） consists of all nilpotent elements of R. As an application, it is also shown that any Gelfand nonsingular extending ring is clean.     

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.     

Nilpotent and invertible values in semiprime rings with generalized derivations

Let R be a semiprime ring and F be a generalized derivation of R and n??? 1 a fixed integer. In this paper we prove the following: (1) If (F(xy) ? yx) ⁿ is either zero or invertible for all ${x,y\in R}$ , then there exists a division ring D such that either R?=?D or R?=?M ₂(D), the 2?× 2 matrix ring. (2) If R is a prime ring and I is a nonzero right ideal of R such that (F(xy) ? yx) ⁿ ?=?0 for all ${x,y \in I}$ , then [I, I]I?=?0, F(x)?=?ax?+?xb for ${a,b\in R}$ and there exist ${\alpha, \beta \in C}$ , the extended centroid of R, such that (a ? ??)I?=?0 and (b ? ??)I?=?0, moreover ((a?+?b)x ? x)I?=?0 for all ${x\in I}$ .