On the uniqueness of almost prime submodules within cyclic uniserial modules

Any ring with Krull dimension satisfies the ascending chain condition on semiprime ideals. This result does not hold more generally for modules. In particular if Ris the first Weyl algebra over a field of characteristic 0 then there are Artinian R-modules which do not satisfy the ascending chain condition on prime submodules. However, if Ris a ring which satisfies a polynomial identity then any R-module with Krull dimension satisfies the ascending chain condition on prime submodules, and, if Ris left Noethe-rian, also the ascending chain condition on semiprime submodules.

An almost prime submodule is a generalization of prime submodule introduced in 2011 by Khashan. This algebraic structure was brought from an algebraic structure in ring theory, prime ideal, and almost prime ideal. This paper aims to construct similar properties of prime ideal and almost prime ideal from ring theory to module theory. The problem that we want to eliminate is the multiplication operation, which is missing in module theory. We use the definition of module annihilator to bridge the gap. This article gives some properties of the prime submodule and almost prime submodule of CMS module over a principal ideal domain. A CSM module is a module that every cyclic submodule. One of the results is that the idempotent submodule is an almost prime submodule.

Let $R\ $be a commutative ring with $1\neq0$ and $M$ be an $R$-module. Suppose that $S\subseteq R\ $is a multiplicatively closed set of $R.\ $Recently Sevim et al. in \cite{SenArTeKo} introduced the notion of an $S$-prime submodule which is a generalization of a prime submodule and used them to characterize certain classes of rings/modules such as prime submodules, simple modules, torsion free modules,\ $S$-Noetherian modules and etc. Afterwards, in \cite{AnArTeKo}, Anderson et al. defined the concepts of $S$-multiplication modules and $S$-cyclic modules which are $S$-versions of multiplication and cyclic modules and extended many results on multiplication and cyclic modules to $S$-multiplication and $S$-cyclic modules. Here, in this article, we introduce and study $S$-comultiplication modules which are the dual notion of $S$-multiplication module. We also characterize certain classes of rings/modules such as comultiplication modules, $S$-second submodules, $S$-prime ideals and $S$-cyclic modules in terms of $S$-comultiplication modules. Moreover, we prove $S$-version of the dual Nakayama's Lemma.

In previous papers by the author (Dilworth [1, 2])(2) methods were developed for studying the arithmetical properties of Birkhoff lattices, that is, the properties of irreducibles and decompositions into irreducibles. These methods, however, required the assumption of both the ascending and descending chain conditions. In this paper we give a new technique which is applicable in general and which under the assumption of merely the ascending chain condition gives results quite as good as those of the previous work. Now the descending chain condition is equivalent to the requirement that every ideal(3) be principal. Hence if the descending chain condition does not hold we find it convenient to relate the arithmetical properties of the lattice to the structure of its lattice of ideals. Furthermore since the Birkhoff condition itself may lose much of its force if the descending chain condition does not hold, a lattice is defined to be a Birkhoff lattice if every element satisfies the Birkhoff condition(4) in the lattice of ideals. Hence if the descending chain condition holds, this definition reduces to that used in the previous papers. In the lattice of ideals, the existence of sufficient covering ideals to make the Birkhoff conditions effective can be proved.

Modern ring theory began when J.H.M. Wedderburn proved his celebrated classification theorem for finite dimensional semisimple algebras over fields. Twenty years later, E. Noether and E. Artin introduced the Ascending Chain Condition (ACC) and the Descending Chain Condition (DCC) as substitutes for finite dimensionality, and Artin proved the analogue of Wedderburn’s Theorem for general semisimple rings. The Wedderburn-Artin theory has since become the cornerstone of noncommutative ring theory, so in this first chapter of our book, it is only fitting that we devote ourselves to an exposition of this basic theory.

Modern ring theory began when J.H.M. Wedderburn proved his celebrated classification theorem for finite dimensional semisimple algebras over fields. Twenty years later, E. Noether and E. Artin introduced the Ascending Chain Condition (ACC) and the Descending Chain Condition (DCC) as substitutes for finite dimensionality, and Artin proved the analogue of Wedderburn’s Theorem for general semisimple rings. The Wedderburn-Artin theory has since become the cornerstone of noncommutative ring theory, so in this first chapter of our book, it is only fitting that we devote ourselves to an exposition of this basic theory.KeywordsLeft IdealDivision RingMatrix RingArtinian RingSimple RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

The polynomial ring over a Goldie ring need not be a Goldie ring

The ascending chain condition (ACC) conjecture for local volumes predicts that the set of local volumes of Kawamata log terminal (klt) singularities x ∈ ( X , Δ ) x\in (X,\Delta ) satisfies the ACC if the coefficients of Δ \Delta belong to a descending chain condition (DCC) set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of δ \delta -plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.

Naturally one wonders whether condition (b) is a consequence of (a) and added strength is given to such a conjecture, if one remembers Hopkins' 3 Theorem to the effect that in rings possessing an identity the ascending chain condition for right-ideals is a consequenice of the descending chain condition. However, it is possible to construct examples of cyclic groups where the descending, though not the ascenidinig, chain conditioni is satisfied by the admissible subgroups. In the light of Hopkins' Theorem, just quoted, it is only natural to assume that there will exist a large class of groups where cyclicity and descending chain condition imply the ascending chain condition; and it is the object of the present note to exhibit such classes of groups. Let A be an abelian group where the composition is written as addition a + b; and assume that A admits the elements in the system M as operators (= right multipliers). The M-subgroup S of A is said to be of length n = n(S), if the M-group S possesses a compositioni series 4 of length n. An ill-subgroup J of A is termed minimal, if Jt# 0 and if J does not

If I is a right ideal of a ring R, I is said to be an annihilator right ideal provided that there is a subset S in R such thatI is said to be injective if it is injective as a submodule of the right regular R-module RR. The purpose of this note is to prove that a prime ring R (not necessarily with 1) which satisfies the ascending chain condition on annihilator right ideals is a simple ring with descending chain condition on one sided ideals if R contains a nonzero right ideal which is injective.

4 - The Structure of Semisimple Rings

In this paper, we study ace (ascending chain condition) and dec (descending chain condition) on different types of subgroups of LCA (locally compact abelian) groups, such as open subgroups, compact subgroups, discrete subgroups, metrizable subgroups, closed divisible subgroups, proper dense subgroups. We characterize compactly generated LCA groups as the class of LCA groups whose open subgroups satisfy ace; the compactly cogenerated groups as the class of LCA groups whose discrete subgroups satisfy dec. We also show that ace and dec on the following classes of subgroups are pairwise equivalent: (a) closed subgroups (b) closed totally disconnected subgroups (c) closed (7-compact subgroups (d) closed metrizable subgroups. We also obtain a characterizati on of those LCA groups which contain no proper dense subgroups.

In this note we investigate Lie algebras which satisfy the descending chain condition on ideals of ideals. We show that a Lie algebra L satisfies this descending chain condition if and only if the following two conditions hold: (i) L contains a finite dimensional solvable ideal N such that every solvable ideal of L is contained in N, and (ii) L/N is a subdirect sum of a finite number of prime algebras satisfying the descending chain condition. We also show that if L is a prime algebra with this chain condition then there exists a Lie algebra B, which is either simple or the tensor product of a simple Lie algebra with a truncated polynomial algebra, such that L is isomorphic to a subalgebra of Der B containing adB. A decade ago a theory of Jordan algebras with descending chain condition on inner ideals was developed [3, Chapter IV] which emulates and connects with the theory of Artinian rings. More recently Benkart [1] studied Lie algebras with descending chain condition on inner ideals (a subspace B of a Lie algebra L is called an inner ideal of L if [B, [B, L]] £ JS). It has not been settled yet whether a Jordan algebra with DCG on inner ideals necessarily has a nilpotent radical. One of the purposes of the present paper is to show that a Lie algebra with DCC on inner ideals has a radical which is solvable and finite dimensional. This follows from the results stated in the last paragraph since any ideal of an ideal is an inner ideal and hence DCC on inner ideals implies DCC on ideals of ideals. It is known that a finite dimensional semisimple Lie algebra M of characteristi c p is not necessarily a direct sum of simple algebras, but there do not seem to be any results published which express M in terms of algebras which belong to a more restricted class than M. A second purpose of this paper is to show that M is a subdirect sum of prime algebras. Rather than finite dimensionality the assumption of DCC on ideals of ideals seems to be the most natural level of generality for this proof. The results in this paper hold for Lie algebras over a field Φ of any characteristic including 2. Suppose now that L is a Lie algebra with DCC on ideals of ideals. We begin with LEMMA 1. If G is a solvable ideal of L, then C is finite

This paper introduces the concept of graded Jgr-2-absorbing primary submodules, a new intermediate structure in graded module theory. Motivated by the need to extend and unify classical notions such as graded prime and graded primary submodules, the study develops a comprehensive framework describing their defining characteristics and relationships. Through a sequence of rigorous theorems and illustrative examples, we establish fundamental properties, equivalence conditions, and inclusion relations that clarify the behavior of these submodules within graded modules. The findings show that graded Jgr-2-absorbing primary submodules generalize several known structures while maintaining distinctive algebraic flexibility. Moreover, the results concerning graded homomorphisms and multiplication modules demonstrate the robustness of the concept and its potential applications in graded algebra. Overall, this work deepens the theoretical understanding of graded algebraic systems and provides a foundation for further research in module and ring theory.