SELF-SMALL ABELIAN GROUPS

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.

A module [Formula: see text] is called coseparable ([Formula: see text]-coseparable) if for every submodule [Formula: see text] of [Formula: see text] such that [Formula: see text] is finitely generated ([Formula: see text] is simple), there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is finitely generated. In this paper, we show that free modules are coseparable. We also investigate whether or not the ([Formula: see text]-)coseparability is stable under taking submodules, factor modules, direct summands, direct sums and direct products. We show that a finite direct sum of coseparable modules is not, in general, coseparable. But the class of [Formula: see text]-coseparable modules is closed under finite direct sums. Moreover, it is shown that the class of coseparable modules over noetherian rings is closed under finite direct sums. A characterization of coseparable modules over noetherian rings is provided. It is also shown that every lifting (H-supplemented) module is coseparable ([Formula: see text]-coseparable).

In this paper, we introduce and study the properties of chain conditions on M-cyclic submodules defined as M-Noetherian and M-Artinian modules and rings. We prove that the property of being iso-M-Noetherian (iso-M-Artinian) is inherited by submodules, quotient modules, and finite sums as well as finite direct sums. Finally, we prove the Hilbert basis theorem for iso-M-Noetherian (iso-M-Artinian) and verify that the property of modules of being M-Noetherian (M-Artinian) and iso-M-Noetherian (iso-M-Artinian) are Morita invariant.

This note investigates torsion-free abelian groups G of finite rank which embed, as subgroups of finite index, in a finite direct sum C of subgroups of the additive group of rational numbers. Specifically, we examine the relationship between G and C when the index of G in C is minimal. Some properties of Warfield duality are developed and used (in the case that G is locally free) to relate our results to earlier ones by Burkhardt and Lady.

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q g of Q by g, for every g ∈ G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G r (r ≥ 1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ⊕ n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.

A module M is called (cofinitely) Rad-⊕-supplemented if every (cofinite) submodule of M has a Rad-supplement that is a direct summand of M. We prove that if M is a coatomic cofinitely Rad-⊕-supplemented module, then M is an irredundant sum of local direct summands. We show that the classes of cofinitely Rad-⊕-supplemented modules and Rad-⊕-supplemented modules are closed under finite direct sums. We also show that every direct summand of a weak duo (cofinitely) Rad-⊕-supplemented module is (cofinitely) Rad-⊕-supplemented.

On duality theory and AB5 ∗ modules

Let X = G/K be a Riemannian symmetric space of the noncompact type and restricted root system BC2 or C2 (except for G = SO0(p, 2) with p > 2 odd). The analysis of the meromorphic continuation of the resolvent of the Laplacian of X is reduced from the analysis of the same problem for a direct product of two isomorphic rank-one Riemannian symmetric spaces of the noncompact type which are not isomorphic to real hyperbolic spaces. We prove that the resolvent of the Laplacian of X can be lifted to a meromorphic function on a Riemann surface which is a branched covering of the complex plane. Its poles, that is the resonances of the Laplacian, are explicitly located on this Riemann surface. The residue operators at the resonances have finite rank. Their images are finite direct sums of finite-dimensional irreducible spherical representations of G.

Classes of operators determined by ordinal indices

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠K* and A is infinite. There existence is also unknown.In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and1. K = A − A,2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.If K is finite this is equivalent to 1 and2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.Finite sharp-fields are special cases of difference sets [De]

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._

Poor modules with no proper poor direct summands

The group D(P) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P. We also obtain partial results on its torsion subgroup. We determine next the structure of Q\otimes D(-) viewed as a functor, which turns out to be a simple functor S_{E,Q}, indexed by the elementary group E of order p^2 and the trivial Out(E)-module Q. Finally we describe a rather strange exact sequence relating Q\otimes D(P), Q\otimes B(P), and Q\otimes R(P), where B(P) is the Burnside ring and R(P) is the Grothendieck ring of QP-modules.

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

We study when some graded rings are Jacobson rings. In particular we obtain that rings strongly graded either by a polycyclic-by-finite group, or by an abelian group of finite torsion free rank are Jacobson rings in case the identity component is a left Noetherian Jacobson ring. For monoid rings R[S] of abelian monoids of finite rank the same result holds without R being left Noetherian.