Linearly minimal Lie algebras

We prove that each nonabelian linearly minimal Lie algebra contains an infinite locally finite linearly minimal subalgebra.     

Commutative algebras of minimal rank

If $\text{A}$ is a finite-dimensional associative k-algebra with unit, then its rank R($\text{A}$), i.e. its bilinear complexity, is never less than $\text{2dim}\text{A}\text{?\#}\text{M}\text{(}\text{A}\text{)}$. $\text{A}$ is said to be of minimal rank if $\text{R(}\text{A}\text{)=2dim}\text{A}\text{?\#}\text{M}\text{(}\text{A}\text{)}$. In this paper we determine for infinite perfect fields k all commutative k-algebras of minimal rank. Roughly speaking, these algebras are built up from simply generated structures which annihilate each other. Furthermore, we indicate how this result can be used to obtain new lower bounds for the rank of specific commutative algebras.     

Rees algebras with minimal multiplicity

Let (R,m) be a local ring. Let S_M denote the Rees algebra S=R[m^rt] localized at its unique maximal homogeneous ideal M=(m,m^rt). Let TN denote the extended Rees algebra T= R[m^rt, t^-1] localized at its unique maximal homogeneous idea N= (t^?1,m,m^r). Multiplicity formulas are developedfor S_M and T_N. These are used to find necessaIy and sufficient conditions on a Cohen-Macaulay local ring (R,m) and r so that S_M and T_N are Cohen-Macaulay with minimal multiplicity     

Buchsbaum homogeneous algebras with minimal multiplicity

In this paper we first give a lower bound on multiplicities for Buchsbaum homogeneous k-algebras A in terms of the dimension d, the codimension c, the initial degree q, and the length of the local cohomology modules of A. Next, we introduce the notion of Buchsbaum k-algebras with minimal multiplicity of degree q, and give several characterizations for those rings. In particular, we will show that those algebras have linear free resolutions. Further, we will give many examples of those algebras.     

Forcing minimal extensions of Boolean algebras

We employ a forcing approach to extending Boolean algebras. A link between some forcings and some cardinal functions on Boolean algebras is found and exploited. We find the following applications:

1) We make Fedorchuk's method more flexible, obtaining, for every cardinal of uncountable cofinality, a consistent example of a Boolean algebra whose every infinite homomorphic image is of cardinality and has a countable dense subalgebra (i.e., its Stone space is a compact S-space whose every infinite closed subspace has weight ). In particular this construction shows that it is consistent that the minimal character of a nonprincipal ultrafilter in a homomorphic image of an algebra can be strictly less than the minimal size of a homomorphic image of , answering a question of J. D. Monk.

2) We prove that for every cardinal of uncountable cofinality it is consistent that and both and exist.

3) By combining these algebras we obtain many examples that answer questions of J.D. Monk.

4) We prove the consistency of MA + CH + there is a countably tight compact space without a point of countable character, complementing results of A. Dow, V. Malykhin, and I. Juhasz. Although the algebra of clopen sets of the above space has no ultrafilter which is countably generated, it is a subalgebra of an algebra all of whose ultrafilters are countably generated. This proves, answering a question of Arhangelskii, that it is consistent that there is a first countable compact space which has a continuous image without a point of countable character.

5) We prove that for any cardinal of uncountable cofinality it is consistent that there is a countably tight Boolean algebra with a distinguished ultrafilter such that for every the algebra is countable and has hereditary character .

     

Term algebras of elementarily equivalent atom structures

We exhibit two relation algebra atom structures such that they are elementarily equivalent but their term algebras are not. This answers Problem 14.19 in Hirsch and Hodkinson’s text Relation Algebras by Games.     

Slice modules over minimal 2-fundamental algebras

We consider a class of algebras whose Auslander-Reiten quivers have starting components that are not generalized standard. For these components we introduce a generalization of a slice and show that only in finitely many cases (up to isomorphism) a slice module is a tilting module. The first named author was supported by the Polish Scientific Grant KBN No 1 P03A 018 27     

Maximal and minimal prime ideals of incidence algebras

Abstract

In this paper we study the relationship between some homological properties such as the weak and the global dimension, the strong n-coherence, and the (n, d)-property, where n and d are two integers, of a commutative ring and its subrings retract. A special application is the transfer of these properties from a commutative ring to its fixed subring with respect to a subgroup of its group of automorphisms. It concludes with a discussion of the scopes and limits of our results.