Modules and Spectral Spaces

We establish conditions for Spec(M) to be Noetherian and spectral space, w.r.t. different topologies. We used rings with Noetherian spectrum to produce plentiful examples of modules with Noetherian spectrum that have not appeared in the literature previously. In particular, we show that every ?-module has Noetherian spectrum. Another main subject of this article is presenting the conditions under which a module is top. In particular, we show that every distributive module is top, every content weak multiplication R-module M is also top, and moreover, if R has Noetherian spectrum, then Spec(M) is a spectral space.     

Laskerian Lattices

In this paper we investigate prime divisors, B _w-primes and zs-primes in C-lattices. Using them some new characterizations are given for compactly packed lattices. Next, we study Noetherian lattices and Laskerian lattices and characterize Laskerian lattices in terms of compactly packed lattices.     

Endomorphism rings and automorphism groups of almost completely decomposable groups

To every non-cuspidal K-rational point on the modular curve X_l(n) a non-commutative Noetherian domain of global dimension 3 can be associated : the Sklyanin algebra. In this paper we give the defining equations of the Sklyanin algebras and their centers when X_l(n) is rational, i.e. n <= 10 or n = 12.     

Cardinalities of Residue Fields of Noetherian Integral Domains

We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ?₁ with a finite residue field, but the statement “There is a Noetherian domain of cardinality ?₂ with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.     

A Notion of Rank for Right Congruences on Semigroups

ABSTRACT

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (?, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian.

Our interest in ranked semigroups stems from the study of the class ± b? _S of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory T _S of ± b? _S , has the model theoretic property of being stable. Moreover, T _S is superstable if and only if S is weakly right Noetherian. In the present article, we show that T _S satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.     

Lie Superalgebras Over 𝔤𝔩2

Whereas Holm proved that the ring of differential operators on a generic hyperplane arrangement is finitely generated as an algebra, the problem of its Noetherian properties is still open. In this article, after proving that the ring of differential operators on a central arrangement is right Noetherian if and only if it is left Noetherian, we prove that the ring of differential operators on a central 2-arrangement is Noetherian. In addition, we prove that its graded ring associated to the order filtration is not Noetherian when the number of the consistuent hyperplanes is greater than 1.     

Semiperfect Prüfer rings and FPF rings

The Asano-Michler theorem states that a 2-sided order R in a simple Artinian ringO is hereditary provided thatR satisfies the three requirements: (AM1) Noetherian; (AM2) nonzero ideals are invertible; (AM3) bounded. We generalize this in one direction by specializing to a semiperfect bounded orderR, and prove thatR is semihereditary assuming only that finitely generated nonzero ideals are invertible (=R is Prüfer). In this case,R ≈ a fulln ×n matrix ringD _n over a valuation domainD. More generally, we study a ringR, called right FPF, over which finitely generated faithful right modules generate the category mod-R of all rightR-modules. We completely determine all semiperfect Noetherian FPF rings: they are finite products of semiperfect Dedekind prime rings and Quasi-Frobenius rings. (For semiprime right FPF rings, we do not require the Noetherian or semiperfect hypothesis in order to obtain a decom-position into prime rings: the acc on direct summands suffices. The “theorem” with “semiperfect” delected is an open problem.     

Noether环的几点注记

本文利用特征模给出了Ｎｏｅｔｈｅｒ环的一些充分必要条件，并由此讨论了Ｎｏｅｔｈｅｒ环上的内射模与平坦模的一些性质     

Noether模的Artin根

本文定义了Ｎｏｃｔｈｅｒ模的Ａｒｔｉｎ根，给出了Ｎｏｃｔｈｅｒ半局部环上有限生成模的Ａｒｔｉｎ根的刻划．最后，作为Ａｒｔｉｎ根理论的应用，给出了一个关于Ｃｏｈｅｎ－Ｍａｃａｕｌａｙ环的定理及一个模的用ｄｅｐｔｈ和ｋｒｕｌｌ维数表示的公式。     

The property of being equationally Noetherian for some soluble groups

Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A₁ ⩾ A₂ ⩾ … A_n ⩾ … such that ⋂A_n = 1 and all factors A_n/A_n+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 46–59, January–February, 2007.     

Properties of cellular classes of chain complexes

In this paper we prove certain properties of cellular and acyclic classes of chain complexes of modules over a commutative Noetherian ring. In particular, we show that if X is finite and belongs to some cellular class C, then Σ ⁿ H _n X also belongs to C, for every n.     

Endomorphism Rings

In this article we construct an affine prime PI ring with a cyclic module whose endomorphism ring is not Jacobson. Then, using similar techniques, we construct an affine prime Noetherian PI ring with a cyclic module whose endomorphism ring is not Noetherian.     

Regular sequences in Rees and symmetric algebras I

In this paper we describe necessary and sufficient conditions for a system of elements a₁,...,a_t of a local Noetherian ring A such that the sequence a₁T,a₁–a₂T,...,a_t–1– a_tT, a_tin the Rees algebra A[a₁T,...,a_tT], T is an indeterminate, constitutes a regular sequence.     

On Bernstein Algebras Satisfying Chain Conditions

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.     

On Bernstein Algebras Satisfying Chain Conditions

Our aim in this article is to study Noetherian and Artinian Bernstein algebras. We show that for Bernstein algebras which are either Jordan or nuclear, each of the Noetherian and Artinian conditions implies finite dimensionality. This result fails for general Noetherian or Artinian Bernstein algebras. We also investigate the relationships between the three finiteness conditions: Noetherian, Artinian, and finitely generated. Especially, we prove that Noetherian Bernstein algebras are finitely generated.     

投射根和亚投射模

本文研究了投射根和亚投射模的性质.利用Noether环、Noether模的性质给出了亚投射模、投射根是投射的条件,环的投射根为零、幂等的条件.最后,给出了一些例子.本文的大部分结果是陈焕艮等人的一系列主要结果的推广.     

On one class of modules that are close to Noetherian

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C _G (A) = 1, A/C _A (G) is not a Noetherian R-module, but the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.     

Artinian Serial Modules over Commutative (or,Left Noetherian) Rings are at Most One Step Away from Being Noetherian

We introduce and study the concept of dual perfect dimension which is a Krull-like dimension extension of the concept of acc on finitely generated submodules. We observe some basic facts for modules with this dimension, which are similar to the basic properties of modules with Noetherian dimension. For Artinian serial modules, we show that these two dimensions coincide. Consequently, we prove that the Noetherian dimension of non-Noetherian Artinian serial modules over the rings of the title is 1.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.