Rings of Invariant Module Type and Automorphism-Invariant Modules

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements then an indecomposable automorphism-invariant right $R$-module must be quasi-injective. In this paper we show that this result fails to hold if $\mathbb F$ is a field with two elements. Dickson and Fuller had further shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements, then $R$ is of right invariant module type if and only if every indecomposable right $R$-module is automorphism-invariant. We extend the result of Dickson and Fuller to any right artinian ring. A ring $R$ is said to be of right automorphism-invariant type (in short, RAI-type) if every finitely generated indecomposable right $R$-module is automorphism-invariant. In this paper we completely characterize an indecomposable right artinian ring of RAI-type.Comment: To appear in Contemporary Mathematics, Amer. Math. So

Dilute stuffing in the pyrochlore iridate Eu2Ir2O7Eu_2Ir_2O_7

The pyrochlore Eu$_2$Ir$_2$O$_7$ has recently attracted significant attention as a candidate Weyl semimetal. The previous reports on this compound unanimously show a thermally induced metal to insulator (MI) transition, concomitant with antiferromagnetic (AFM) long-range ordering of the Ir-moments below T$_\textit{N} \sim$120 K. However, there are contradictory reports concerning the slope d$\rho/$dT of the resistivity plots ($\rho$) in the "metallic" state above the metal-insulator (MI) transition, and the value of $\rho$ in the insulating state, both of which show significant sample dependence. Here, we explore this issue by investigating six different Eu$_2$Ir$_2$O$_7$ samples with slightly varying Eu:Ir ratio. High-resolution synchrotron powder diffraction are done to probe minor variations in the cell parameters of the various Eu$_2$Ir$_2$O$_7$ samples investigated here. Specific heat (C$_p$) and magnetic susceptibility of all the samples showed long-range antiferromagnetic ordering upon cooling below T$_\textit{N} \sim$120 K. The transitions are, however, found to be smeared out for the off-stoichiometric samples. We show that the sign of d$\rho/$dT above the metal-insulator (MI) transition is highly sensitive to the unit cell length, which, in turn, depends on the level of Eu-stuffing at the Ir-site. Samples with composition close to the ideal stoichiometry (Eu : Ir $=$ 1) showed a change of sign of d$\rho/$dT from negative to positive upon cooling below a certain temperature T $^*$ $>$ T$_\textit{MI}$. With increasing Eu-stuffing T$^*$ decreased until a negative d$\rho/$dT persisted without any sign change down to T$_\textit{MI}$.Comment: 12 pages, 7 figure

Tag-modules with complement submodules H

The concept of a QTAG-module MR was given by Singh [8]. The structure theory of such modules has been developed on similar lines as that of torsion abelian groups. If a module MR is such that M⊕M is a QTAG-module, it is called a strongly TAG-module. This in turn leads to the concept of a primary TAG-module and its periodicity. In the present paper some decomposition theorems for those primary TAG-modules in which all h-neat submodules are h-pure are proved. Unlike torsion abelian groups, there exist primary TAG-modules of infinite periodicities. Such modules are studied in the last section. The results proved in this paper indicate that the structure theory of primary TAG-modules of infinite periodicity is not very similar to that oftorsion abelian groups

Anomalous lattice contraction and emergent electronic phases in Bi-doped Eu2_2Ir2_2O7_7

We study the pyrochlore series (Eu$_{1-x}$Bi$_x$)$_2$Ir$_2$O$_7$ for $0 \leq x \leq 1$. We show that for small $x$, the lattice undergoes an anomalous contraction but the all-in/all-out and metal-to-insulator transitions remain robust, and the resistivity approaches a $1/T$ dependence at low-T, suggesting proximity to the Weyl semimetallic phase, as previously predicted theoretically. At the boundary between Eu$_2$Ir$_2$O$_7$ and Bi$_2$Ir$_2$O$_7$ a qualitatively different ground state emerges, which is characterized by its unusual metallic behavior and absence of magnetic ordering at least down to $0.02$ K.Comment: 5 Pages, 4 figure