Semigroup Graded Rings and Jacobson Rings

We know quite a lot about the general structure of ω-stable solvable centerless groups of finite Morley rank. Abelian groups of finite Morley rank are also well-understood. By comparison, nonabelian nilpotent groups are a mystery except for the following general results:• An ω1-categorical torsion-free nonabelian nilpotent group is an algebraic group over an algebraically closed field of characteristic 0 [Z3].• A nilpotent group of finite Morley rank is the central product of a definable subgroup of finite exponent and of a definable divisible subgroup [N3].• A divisible nilpotent group of finite Morley rank is the direct product of its torsion part (which is central) and of a torsion-free subgroup [N3].However, we do not understand nilpotent groups of bounded exponent. It seems that the classification of nilpotent (but nonabelian) p-groups of finite Morley rank is impossible. Even the nilpotent groups of Morley rank 2 contain insurmountable difficulties [C], [T] . At first glance, this may seem to be an obstacle to proving the Cherlin-Zil'ber conjecture (“simple groups of finite Morley rank are algebraic groups”). Our purpose in this article is to show that if such a group is a definable subgroup of a nonnilpotent group, then it is possible to obtain a classification within the boundaries of our present knowledge. In this respect, our article may be considered as a relief to those who are trying to classify simple groups of finite Morley rank.Before explicitly stating our result, we need the following definition.

Let π be a set of primes. We will call a group π-separated if it can be decomposed as a central product of a π-torsion group of bounded exponent and a π-radicable group. It is easy to see that an abelian π-separated group can in fact be decomposed as a direct product of bounded and π-radicable factors (Lemma 1.1 below). A poly-π-separated group is one which can be obtained from π-separated groups by forming a finite series of group extensions. We will show here:Theorem 1. Every poly-π-separated nilpotent group is π-separated.The need for such a result arises in model theory in the case that π is the set of all primes, in which case we refer simply to separated and poly-separated groups. In connection with the conjecture that ω-stable simple groups are algebraic, it is useful to have a structure theory for solvable ω-stable groups analogous to the theory available in the algebraic case over algebraically closed fields. In particular the structure of nilpotent ω-stable groups is of interest, for example in connection with the known result [Zi1], [Ne] that the derived subgroup of a connected solvable ω-stable group of finite Morley rank is nilpotent.As a model-theoretic application of Theorem 1 we obtain:Theorem 2. If G is an ω-stable nilpotent group then G may be decomposed as a central product B * D with B and D 0-definable subgroups, B torsion of bounded exponent, and D radicable. In particular, B and D are ω-stable. Furthermore, ω · rk(B ∩ D) ≤ rk D.Corollary. The ω-stable groups of finite Morley rank are exactly the central products B * D of ω-stable nilpotent groups of finite Morley rank with B torsion of bounded exponent, D radicable, and B ∩ D finite.In addition to the purely algebraic Theorem 1, these results depend on Macintyre's characterization [Mac] of the ω-stable abelian groups as exactly the separated ones.

Soluble Groups of Type (FP)∞ have Finite Torsion-Free Rank

This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.

Coset-minimal groups

An extension E of an Abelian group A by a locally supersolvable group is studied under the assumption that A has a finite series of subgroups normal in E, each of whose factors satisfies one of the following conditions: 1) min-E; 2) max-E; 3) the factor has finite rank and is torsion-free. It is proved that if E induces in the factors of the series in A hypercyclic groups of automorphisms and A is the locally supersolvable coradical of E, then A is complemented in E and all complements are conjugate in E. An analog of this result for locally nilpotent extensions of Abelian groups is also noted.

We develop a theory of multi-stage degenerations of toric varieties over finite rank valuation rings, extending the Mumford--Gubler theory in rank one. Such degenerations are constructed from fan-like structures over totally ordered abelian groups of finite rank. Our main theorem describes the geometry of successive special fibers in the degeneration in terms of the polyhedral geometry of a system of recession complexes associated to the fan.

It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial.

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._

Properties of abelian groups related to a given finite rank torsion free abelian group A are analyzed in terms of End (A), the endomorphism ring of A. This point of view gives rise to generalizations of some classical theorems by R. Baer and examples of pathological direct sum decompositions of finite rank torsion free abelian groups.

It is shown that the cohomological dimension of an abelian monoid is equal to that of its group reflection provided that the monoid is either finitely generated or cancellative.Let M be a monoid and let bea nonzero ring with identity.We shall denote the monoid ring of M over R by RM.The R-homological dimension of M is defined by hdRM = wd R and, similarly, the R-cohomological dimension of M is defined by cdRM = pd R where R is considered as a trivial Af-module (i.e.x r = r for x M, r R) and where wd, pd denote weak and projective dimension respectively.Clearly hdRM < cdRM and the equality holds when RM is noetherian (e.g. when R is noetherian and M is finitely generated abelian).If there is a ring homomorphism S -> R then cd M < cds M and hdR M < hds M. In particular, cd M < cd M and hd M < hd M where we omit the subscript when R = Z.Recall that a group M is said to have -torsion if there exists an element of M whose order is not invertible in R. In this case cdRM = oo.If M is an Nn-generated abelian group (n > -1) with no -torsion, then hd M = rank M and cd M = n + 1 + rank M (see, for example, [1], [4] and [10]).In this paper we prove the following.Theorem A. // M is an abelian monoid then hdRM = hdRM.Theorem B. // M is an abelian monoid which is either cancellative or finitely generated, then cd M = cd M.Here M denotes the group reflection of M.An example of Nico [8] shows that Theorem B does not hold for arbitrary abelian monoids.1. Homological dimension of an abelian monoid.The group reflection M of a monoid M is defined to be the image of M under the left adjoint of the forgetful functor from Groups to Monoids.It is not hard to show that M is isomorphic to the quotient of the free group of M by the normal subgroup generated by elements

We study the unification problem for a class of equational theories that comprises important examples like abelian monoids (AC), idempotent abelian monoids (ACI), and abelian groups. Monoidal theories have the common characteristic that unification algorithms are based on solving linear equation systems over a semiring.The close correspondence between unification and linear algebra can be used to characterize the unification type of monoidal theories in purely algebraic terms, and an application of Hilbert's Basis Theorem gives a sufficient criterion for a monoidal theory to be unitary.

Let $A$ be a finite rank torsion--free abelian group. Then there exist direct decompositions $A=B\oplus C$ where $B$ is completely decomposable and $C$ has no rank 1 direct summand. In such a decomposition $B$ is unique up to isomorphism and $C$ unique up to near-isomorphism.

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups