C. Vinsonhaler

Large E-rings exist

A ring R with 1 is called an E-ring provided Hom,(R, R) z R under the map cp + 1~. The class of E-rings was defined and studied by Schultz [S] in 1973, and further investigated by Bowshell and Schultz [BS] in 1977. Examples of E-rings include Z/(n), subrings of Q, and pure subrings of the ring of p-adic integers. More interesting examples are the torsion-free E-rings of finite rank. These were characterized in [BS] as those rings quasi-isomorphic to R, x R, x . . . x R,, where each Ri is a strongly indecomposable subring of an algebraic number field and Hom,(R,, Rj) = 0 for i #j. In spite of their seemingly specialized nature, such rings have played an important role in the theory of torsion-free groups of finite rank, dating back to Beaumont-Pierce [BPl, BP2], and Pierce [P] in 1960-1961. (See also [APRVW, NR, PV, RI.) Relatively little has been published on infinite rank torsion-free E-rings. Indeed, until recently, the only examples of these were provided by the pure subrings of the p-adic integers. In this paper, the “Black Box” of Shelah is used to construct a host of large E-rings. We show that any torsion-free p-reduced, p-cotorsion-free, commutative ring S may be embed88 0021-8693/87

Dualities for torsion-free abelian groups of finite rank

3.00

Isomorphism invariants for abelian groups

Two well-known dualities have been very useful in the study of torsionfree abelian groups of finite rank: Warlield duality for locally free groups and Arnold duality for quotient divisible groups [Wa, Ar]. In this note we establish a duality, on classes of torsion-free abelian groups of finite rank, which generalizes both Warlield and Arnold duality. Our results were inspired by a recent paper of Fomin [FOG], who constructed a duality which is also a special case of the one presented here. The basic idea is to write a torsion-free abelian group of finite rank (hereafter, “group”) G as a sum G = G, + G,, where G, is locally free and G, is quotient divisible. A dual for G is then obtained by adding the War-field dual of G,, W(G,), and the Arnold dual of Gz, A(G,) (inside of Hom(G, Q)). The idea of breaking up G into a locally free and a quotient divisible part is not a new one-it was investigated by Murley in [Mu]. As noted by Murley, one easy way to obtain G, and Gz is to take a full free subgroup F of G and write G/F= D 0 R, where D is a divisible and R is a reduced torsion group. Then choose subgroups G, and G2 of G containing F so that G,IF = R and G,/F = D. In general, the G, obtained in this way is decidedly non-unique, even up to quasi-isomorphism. It is therefore somewhat surprising that we can form G* = W(G,) + A(G,) to obtain a group which, up to quasi-equality, is independent of the choice ofG,. Further complexity can be introduced due to the fact that the Warlield and Arnold duals treat free and divisible localizations differently, prompting our definition of PX-groups (Definition 2.1). On the category of PX-groups and quasi-homomorphisms (in which objects are quasi-equality classes), G + G* defines an exact contravariant functor (Theorem 2.13). The PX-group G* has a special property, which was also studied by Murley. On the PX-groups with this special property, the “dualizable PX-groups,” * determines a duality (Theorem 3.3), which generalizes the dualities of Warlield, Arnold, and Fomin (Proposition 3.5). We remark 474 OO21-8693/90

Multivariate Analysis of Pension Plan Mortality Data

3.00

A Class of B(2)-Groups

Let A=(A 1 ,...,A n ) be an n-tuple of subgroups of the additive group, Q, of rational numbers and let G(A) be the kernel of the summation map A 1 ○+...○+A n →∑A i and G[A] the cokernel of the digonal embedding ∩A i →A 1 ○+...○+A n . A complete set of isomorphism invariants for all strongly indecomposable abelian groups of the form G(A), respectively, G[A], is given. These invariants are then extended to complete sets of isomorphism invariants for direct sums of such groups and for a class of mixed abelian groups properly containing the class of Warfield groups

Separable torsion-free abelian E∗-groups

Abstract This paper uses the logistic regression model to examine private pension plan data for 1989–95 collected by the Retirement Plans Experience Committee of the Society of Actuaries. When only one explanatory variable, such as annuity class size, is used in modeling mortality rates, the model provides a reasonable fit to the data. Multiple explanatory variables give less satisfactory results.

Regulating hulls of almost completely decomposable groups

Abstract We consider a subclass of Butler groups, the OB (2) -groups. The class B (2) consists of the groups G which appear in an exact sequence 0 → K → D → G → 0 where D is a finite rank completely decomposable group and K is a rank two pure subgroup of D . The groups G belonging to OB (2)⊆ B (2) satisfy an additional condition called “overlap”. We discuss three questions: (1) When is G strongly indecomposable? (2) To what extent is D determined by G ? (3) Is there a reasonable set of numerical invariants for such a G ? We find definitive answers for these questions in the case of strongly indecomposable balanced OB (2)-groups. We also obtain partial results without the balanced hypothesis.

Two-sided E-rings

Abstract A ring R is said to be a unique addition ring (UA-ring) if any semigroup isomorphism R ∗ = (R, ∗) ∼- (S, ∗) = S ∗ of multiplicative semigroups with another ring S is always a ring isomorphism. See [5, 7–9] for earlier work on UA-rings. Depending on the context, we may or may not regard 0 as an element of R ∗ . An abelian group G is called a UA-group if its endomorphism ring E ( G ) is a UA-ring. Given an abelian group G , denote by E ∗ (G) the semigroup of all endomorphisms of G and let R G be the collection of all rings R such that R ∗ ∼- E ∗ (G) . The group G is said to be an E ∗ - group if for every ring ( E ∗ (G), ⊕ ), where ⊕ is an addition on the semigroup E ∗ (G) , there is an abelian group H such that ( E ∗ (G), ⊕ ) is (isomorphic to) the endomorphism ring of H . Equivalently, G is an E ∗ - group if for every ring R in R G there is an abelian group H such that R is (isomorphic to) the endomorphism ring of H . Section 1 is a study of separable torsion-free abelian UA-groups. In Section 2 we develop necessary and sufficient conditions for a torsion-free separable group to be an E ∗ - group . All groups are abelian.

Torsion-free modules of finite balanced-projective dimension over valuation domains

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.

Quasi-isomorphism invariants for two classes of finite rank Butler groups

Abstract Let R be a ring, 1∈R, and R+ the additive group of R. We define Mult(R) to be the subring of End(R+) generated by all left and right multiplications by elements of R. The ring R is called a two-sided E-ring if End(R+)=Mult(R). If R is torsion-free of finite rank (tffr), we call R a quasi-two-sided E-ring if Q End(R + )= Q Mult(R) . We investigate (quasi)-two-sided E-rings, give several examples and construct large two-sided E-rings R with prescribed center S such that End(R+)=Mult(R)≈R⊗SRop≈R. Thus our rings R are examples of two-sided E-rings that are weak E-rings as well, i.e. R≈ End(R+), but R is not an E-ring.