On the Content of Polynomials Over Semirings and Its Applications

In this paper, we prove that Dedekind-Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.Comment: Final version published at J. Algebra Appl., one reference added, three minor editorial change

Indecomposable modules and Gelfand rings

It is proved that a commutative ring is clean if and only if it is Gelfand with a totally disconnected maximal spectrum. Commutative rings for which each indecomposable module has a local endomorphism ring are studied. These rings are clean and elementary divisor rings

Type-Decomposition of a Pseudo-Effect Algebra

The theory of direct decomposition of a centrally orthocomplete effect algebra into direct summands of various types utilizes the notion of a type-determining (TD) set. A pseudo-effect algebra (PEA) is a (possibly) noncommutative version of an effect algebra. In this article we develop the basic theory of centrally orthocomplete PEAs, generalize the notion of a TD set to PEAs, and show that TD sets induce decompositions of centrally orthocomplete PEAs into direct summands.Comment: 18 page

Lie algebras and 3-transpositions

We describe a construction of an algebra over the field of order 2 starting from a conjugacy class of 3-transpositions in a group. In particular, we determine which simple Lie algebras arise by this construction. Among other things, this construction yields a natural embedding of the sporadic simple group \Fi{22} in the group $^2E_6(2)$.Comment: 23 page

Twisted K-Theory of Lie Groups

I determine the twisted K-theory of all compact simply connected simple Lie groups. The computation reduces via the Freed-Hopkins-Teleman theorem to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al.Comment: 16 page

The splitting principle for Grothendieck rings of schemes

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/32761/1/0000132.pd

The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres

Generalized classical Albert-Zassenhaus Lie algebras

Two very large classes GCAZ and CAZK of Lie algebras are introduced, which contain all sums of classical, Albert-Zassenhaus, generalized Witt algebras of Kaplansky and associated holomorphs. Their rootsystems R are classified up to isomorphism. The group Aut L of automorphisms of L is shown to contain extensions of the Weyl group of R and the inner automorphism groups of classical Lie algebra complements of the Witt subalgebra of L. The Weyl group extension in Aut L acts transitively by conjugation on the classical complements, under general conditions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25511/1/0000052.pd

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page

Generalised Witt algebras and idealizers

Let $\Bbbk$ be an algebraically closed field of characteristic zero, and let $\Gamma$ be an additive subgroup of $\Bbbk$. Results of Kaplansky-Santharoubane and Su classify intermediate series representations of the generalised Witt algebra $W_\Gamma$ in terms of three families, one parameterised by ${\mathbb A}^2$ and two by ${\mathbb P}^1$. In this note, we use the first family to construct a homomorphism $\Phi$ from the enveloping algebra $U(W_\Gamma)$ to a skew extension of ${\Bbbk}[a,b]$. We show that the image of $\Phi$ is contained in a (double) idealizer subring of this skew extension and that the representation theory of idealizers explains the three families. We further show that the image of $U(W_\Gamma)$ under $\Phi$ is not left or right noetherian, giving a new proof that $U(W_\Gamma)$ is not noetherian. We construct $\Phi$ as an application of a general technique to create ring homomorphisms from shift-invariant families of modules. Let $G$ be an arbitrary group and let $A$ be a $G$-graded ring. A graded $A$-module $M$ is an intermediate series module if $M_g$ is one-dimensional for all $g \in G$. Given a shift-invariant family of intermediate series $A$-modules parametrised by a scheme $X$, we construct a homomorphism $\Phi$ from $A$ to a skew-extension of ${\Bbbk}[X]$. The kernel of $\Phi$ consists of those elements which annihilate all modules in $X$.Comment: 9 pages; to appear in J. Algebr