Fragments with finiteness conditions in particular over group rings

Abstract length functions on groups were introduced by Lyndon [5]. In [2] Chiswell showed that for a group action on an IR-tree there is a length function associated with each point of the tree, and moreover any length function can occur in this way. This paper establishes necessary and sufficient conditions for two length functions to arise from the same action of a group on a tree.     

Ring extensions with some finiteness conditions on the set of intermediate rings

A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ S is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension R ⊆ S to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prüfer and integral domains satisfying the corresponding finiteness conditions.     

Tabor groups with finiteness conditions

We analyse Tabor groups where every element has finite order and we characterise finite Tabor groups.     

Generalized finiteness conditions

For a module M for various quotient modules M/N of M, cardinality restrictions are placed on the number of generators of submodules of M/N, and on the (infinite) Goldie dimensions of M/N. The conditions FGS and TC of R. Kurshan are formulated and generalized for arbitrary cardinals. The module theoretic consequences of these and other such finite-ness conditions with ascending chain conditions are explored.     

Shirshov finiteness over rings of constants

Let L be a finite-dimensional restricted differential Lie C-algebra of R-continuous derivations of a prime ring R of characteristic p>0, with generalized centroid C. We prove that if the associative inner part of L is quasi-Frobenius then R contains a nonzero element a and elements v₁,…,v_n, such that for any x∈R we have the expansion , where are homorphisms of right R^L-modules . This gives rise to a certain relation on a ring over some subring, known as Shirshov local finiteness. The structure of (R, R^L)-subbimodules in a left Martindale ring of quotients is elucidated. Supported by RFFR grant No. 95-01-01356a, and by the CONACYT of Mexico, Catedra Patrimonial 940411. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 219–238, March–April, 1997.     

Nonconstructive negative algebras with finiteness conditions

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 6, pp. 195–198, November–December, 1992.     

On finiteness of chains of intermediate rings

An extension of integral domains R í S{R\subseteq S} is said to have the “finite length of intermediate chains of domains” property (for short FICP) if each chain of intermediate rings between R and S is finite. The main purpose of this paper is to characterize when R í S{R\subseteq S} has FICP in case R ^* (the integral closure of R in S) is a finite dimensional semilocal domain. This generalizes a theorem due to Gilmer, in which S is the quotient field of R. Examples illustrating the sharpness and the limits of our results are settled.     

A commutativity and finiteness condition for rings

We prove that if a ring R has the property that XY = YX for all infinite subsets X and Y, then R is either finite or commutative.     

On finiteness of chains of intermediate rings

An extension of integral domains is said to have the “finite length of intermediate chains of domains” property (for short FICP) if each chain of intermediate rings between R and S is finite. The main purpose of this paper is to characterize when has FICP in case R ^* (the integral closure of R in S) is a finite dimensional semilocal domain. This generalizes a theorem due to Gilmer, in which S is the quotient field of R. Examples illustrating the sharpness and the limits of our results are settled.      

Graded prime Goldie rings

Prime Goldie rings graded by a group are studied. It is proved that there exists a completely gr-reducible graded ring of fractions in the case of a grading group with a finite commutator subgroup (which enhances the results of Goodearl–Stafford obtained for Abelian groups). An example where such ring does not exist is constructed (a counterexample for a semi-prime ring was already known).     

Graded rings and essential ideals

LetG be a group andA aG-graded ring. A (graded) idealI ofA is (graded) essential ifI⊃J≠0 wheneverJ is a nonzero (graded) ideal ofA. In this paper we study the relationship between graded essential ideals ofA, essential ideals of the identity componentA _e and essential ideals of the smash productA#G ^*. We apply our results to prime essential rings, irredundant subdirect sums and essentially nilpotent rings.