Related papers: Unifying some classical results on Artinian rings …

We introduce the notion of Artinian meadow as an algebraic structure constructed from an Artinian ring which is also a common meadow, i.e.\ a commutative and associative structure with two operations (addition and multiplication) with…

Injective, pure-injective and fp-injective modules are well known to provide for approximations in the category Mod-R for an arbitrary ring R. We prove that this fails for many other generalizations of injectivity: the $C_1$, $C_2$, $C_3$,…

Let $R$ be a commutative unital ring, $a\in R$ and $t$ a positive integer. $a^{t}$-reduced $R$-modules and universally $a^{t}$-reduced $R$-modules are defined and their properties given. Known (resp. new) results about reduced $R$-modules…

The notion of trivial extension of a ring by a module has been extensively studied and used in ring theory as well as in various other areas of research like cohomology theory, representation theory, category theory and homological algebra.…

Silting modules are abundant. Indeed, they parametrise the definable torsion classes over a noetherian ring, and the hereditary torsion pairs of finite type over a commutative ring. Also the universal localisations of a hereditary ring, or…

Let $R$ be a commutative unital ring, $\mathfrak{ a}$ an ideal of $R$ and $M$ a fixed $R$-module. We introduce and study generalisations of $\mathfrak{a}$-reduced modules, $\mathfrak{R}_{\mathfrak{ a}}$ and $\mathfrak{a}$-coreduced modules,…

This paper provides a new approach to proving generalizations of the F.&M. Riesz Theorem, for example, the result of Helson and Lowdenslager, the result of Forelli (and de Leeuw and Glicksberg), and more recent results of Yamagushi. We…

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective…

P\'olya's Positivstellensatz and Handelman's Positivstellensatz are known to be concrete instances of the abstract Archimedean Representation Theorem for (commutative unital) rings. We generalise the Archimedean Representation Theorem to…

We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the $\mu$-ordinary locus (with an…

Let $p$ be a prime and $\mathbb{Z}_p$ the ring of $p$-adic integers. Let $G$ denote the simple group of order 168, 504 or 360. In this paper, we study the structure of the $\chi$-part of a $\mathbb{Z}_p[\mathrm{Im}(\chi)][G]$-module come…

Measure and integral are two closely related, but distinct objects of study. Nonetheless, they are both real-valued lattice valuations: order preserving real-valued functions $\phi$ on a lattice $L$ which are modular, i.e.,…

In this paper, using $p$-adic analysis and $p$-adic L-functions, we show how to extend classical congruences (due to Wilson, Gauss, Dirichlet, Jacobi, Wolstenholme, Glaisher, Morley, Lemher and other people) to modulo $p^k$ for any $k>0$.

The Mitsch order is already known as a natural partial order for semigroups and rings. The purpose of this paper is to further study of the Mitsch order on modules by investigating basic properties via endomorphism rings. And so this study…

We study graduated orders over completed group rings of $1$-dimensional admissible $p$-adic Lie groups, and verify the equivariant $p$-adic Artin conjecture for such orders. Following Jacobinski and Plesken, we obtain a formula for the…

Let $A$ be a ring with minimum condition on principal right ideals. It is proved that $\aleph_0$-distributive right (left) $A$-modules coincide with Artinian (Noetherian) right (left) $A$-modules. Rings, over which all right modules are…

Let $R$ be a commutative Noetherian ring. It is shown that $R$ is Artinian if and only if every $R$-module is good, if and only if every $R$-module is representable. As a result, it follows that every nonzero submodule of any representable…

The present work develops a framework to derive piecewise polynomial measures arising from invariant measures on adjoint orbits in the context of compact and semisimple Lie groups. These measures are computed from orbital integrals via…

Let $R$ be a commutative Noetherian ring and $M$ be a finitely generated $R$-module. Considering the new concept of linkage of ideals over a module, we study associated prime ideals, cofiniteness and Artinianness of local cohomology modules…

We survey noetherian rings $A$ over which the injective hull of every simple module is locally artinian. Then we give a general construction for algebras $A$ that do not have this property. In characteristic 0, we also complete the…