Toma Albu

LOCALIZATION OF MODULAR LATTICES, KRULL DIMENSION, AND THE HOPKINS-LEVITZKI THEOREM. II

The aim of this note is to correct some mistakes in the paper mentioned in the title. An unpublished example of M.L. Teply is given of a non-Noetherian commutative domain which has Krull dimension 2 and zero Jacobson radical.

CHAIN CONDITIONS ON QUOTIENT FINITE DIMENSIONAL MODULES

This paper is motivated by a recent work of C. Faith [9] who has proved that a quotient finite dimensional module which satisfies the ascending chain condition on subdirectly irreducible submodules is Noetherian. A natural question to ask (also raised by Faith [11]) is whether its dual holds true. We answer this in the affirmative, and provide a characterization of Artinian modules dual to Faiths Theorem. We also extend these results to the more general settings of dual Krull dimension and Krull dimension respectively.

Localization of modular lattices, Krull dimension, and the Hopkins-Levitzki theorem (I)

The Hopkins–Levitzki Theorem, discovered independently in 1939 by C. Hopkins and J. Levitzki states that a right Artinian ring with identity is right Noetherian. In the 1970s and 1980s it has been generalized to modules over non-unital rings by Shock[10], to modules satisfying the descending chain condition relative to a heriditary torsion theory by Miller-Teply[7], to Grothendieck categories by Năstăsescu [8], and to upper continuous modular lattices by Albu [1]. The importance of the relative Hopkins-Levitzki Theorem in investigating the structure of some relevant classes of modules, including injectives as well as projectives is revealed in [3] and [6], where the main body of both these monographs deals with this topic. A discussion on the various forms of the Hopkins–Levitzki Theorem for modules and Grothendieck categories and the connection between them may be found in [3].

INFINITE FIELD EXTENSIONS WITH COGALOIS CORRESPONDENCE

ABSTRACT The aim of this paper is to generalize a series of results by T. Albu and M. T¸ena[6] from infinite bounded –Cogalois extensions to arbitrary infinite –Cogalois extensions.

SOME EXAMPLES IN COGALOIS THEORY WITH APPLICATIONS TO ELEMENTARY FIELD ARITHMETIC

The aim of this paper is to provide some examples in Cogalois Theory showing that the property of a field extension to be radical (resp. Kneser, or Cogalois) is not transitive and is not inherited by subextensions. Our examples refer especially to extensions of type . We also effectively calculate the Cogalois groups of these extensions. A series of applications to elementary arithmetic of fields, like: • for what n, d ∈ ℕ* is a sum of radicals of positive rational numbers • when is a finite sum of monomials of form , where r, j1,…, jr ∈ ℕ*, c ∈ ℚ*, and are also presented.

Generalized deviation of posets and modular lattices

Abstract The existence and relationship of the Γ-deviation and dual Γ-deviation of a poset is investigated, where Γ is a nonempty set of linear order types. If Γ is a finite set of linear order types, then bounds are given for the Γ-deviation of a poset (when it exists). If Γ is an arbitrary set of indecomposable order types and L is a modular lattice, then the Γ-deviation behaves in a classical manner on subintervals of L. Examples are given to illustrate the results.

Dual Relative Krull Dimension of Modules Over Commutative Rings

The aim of this paper is to study the relationship between the dual Krull dimension of R-modules relative to a Gabriel topology F on a commutative ring R and the Krull dimension of R relative to F.

Kummer extensions with few roots of unity

Let K be an arbitrary field, K an algebraic closure of K, n ≥ 1 a natural number, and μn(K) = {z|z ∈ K,zn = 1}. A finite Kummer extension of K of exponent n with few (resp., many) roots of unity is an extension K(x1,…,xk) of K, where k ∈ N∗, x1, …, xk ∈ K∗ are such that xin ∈ K for all i, 1 ≤ i ≤ k, and μn(K) ∩ K(x1, …, xk) ⊆ {1, −1} (resp., μn(K) ⊆ K). We prove that a classical result concerning the evaluation of the degree [K(x1, …, xk):K] holds equally for finite Kummer extensions of exponent n with few or with many roots of unity, if Char(K) ∤ n. For such an extension K ⊆ K(xi, …, xk) for which [K(x1, …, xk) : K] = Π1 ≤ i ≤ k[K(xi) : K], it is shown that K(x1, …, xk) = K(x1 + ctdot; + xk). Further, if K is an arbitrary field and n is a prime number other than Char(K), then any extension K ⊆ K(x1, …, xk), where k ∈ N∗ and x1, …, xk ∈ K∗ are such that xin ∈ K for all i, 1 ≤ i ≤ k, is a finite Kummer extension of exponent n with few or with many roots of unity, and, consequently, the above results hold in this case. Our results complete, unify, or extend some of the results of J. L. Mordell, H. Hasse, A. Baker and H. M. Stark, I. Kaplansky, I. Richards, and H. D. Ursell appearing in the literature, and reveal the connections between them.

On commutative grothendieck categories having a Noetherian cogenerator

I t is well-known the following theorem, discovered independently in 1939 by C. Hopkins [11] and J. Levitzki [12]: any right Artinian ring with identity element is right l~oetherian. Some a t tempts were made in the last years with a view to generalize this theorem to arbi trary Grothendieck categories. Thus, in 1969 J. E. Roos [16] gave an example of a locally Artinian Grothendieek category which is not locally I~oetherian. However, if /~ is a commutat ive ring with unit element and ~ is a quotient category of the category Mod-R of R-modules by an arbi trary localizing subcategory Yof Mod-~ (i. e. ~ is a commutat ive Grothendieck category), it was proved in [3; 4.7] tha t if T(R) is an Artinian object in ~ then T(R) is also a Noetherian object, where T: Mod-R--> Mod-R/Yis the canonical functor. The following problem was raised also in [3; 4.8]: does this result hold for a noncommuta t ive ring /~ with unit element ? This question was solved affirmatively by M. L. Teply and R. W. Miller [20 ; 1.4]. A very short and elegant proof of the result of Teply and Miller was given by C. N~st~sescu [14; 1.3], who proved the following more general theorem: if ~ is an arbi trary Grothendieek category which has an Artinian generator U, then U is l~oetherian. This s ta tement seems to be the most natural way to place the Hopkins-Levitzki theorem in the general setting of Grothendieck categories. The aim of the present paper is to s tudy the dual situation from the HopkinsLevitzki theorem in Ns version, i.e. the case of a Grothendieck category having a Noetherian cogenerator. Thus, for a commutative Grothendieck category we prove tha t ~ has a l~oetherian cogenerator if and only if ~ has an Artinian generator. As a corollary we obtain that for a commutat ive Grothendieck category holds the dual of Hopkins-Levitzki theorem: each Iqoetherian cogenerator of ~ is Artinian. We end the paper with a list of open questions. I wish to thank Dr. C. Ns for many stimulating conversations.

Infinite group-graded rings, rings of endomorphisms, and localization

The paper is divided into four sections. The first section gives a matricial description R ▿ G of the ring EndR-gr(U), which contains as a unital subring the smash product R#G, R= ⊕x ∈ GRx being a group-graded ring and =⊕x ∈ GR(x) the canonical generator of the category R-gr. Section 2 proves mainly the following two facts: for each subring B of R ▿ G containing R#G, R-gr is equivalent to a quotient category of B-mod, and the ring R ▿ G is isomorphic to a ring of quotients of B. Section 3 investigates the structure of graded endomorphism rings of the type ENDR(⊕x ∈ GM(x)), where M∈R-gr is a finitely generated R-module. In the last section it is shown that R-mod is equivalent to a quotient category of the category ENDR(U)-mod, and EndR(U) is isomorphic to a ring of quotients of the ring ENDR(U).