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In this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results. [ABSTRACT FROM AUTHOR]

We prove a Noetherian criterion for a sequence of modules with linear maps between them. This generalizes a Noetherian criterion of Gan and Li for infinite EI categories. We apply our criterion to the linear categories associated to certain diagram algebras defined by Patzt. [ABSTRACT FROM AUTHOR]

AbstractThe centrally extended Heisenberg double of quantum SL2 is a smash product of the quantized coordinate ring Oq(M2) of 2×2 matrices and the Hopf algebra Uq(sl2). For this algebra, we give a classification of all simple weight modules with a finite-dimensional weight space, as well as the so-called Uq(sl2)-Harish-Chandra modules. [ABSTRACT FROM AUTHOR]

Under semi-weak and weak compatibility conditions of bimodules, we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts. Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules. [ABSTRACT FROM AUTHOR]

It is well known that the ring radical theory can be approached via language of modules. In this work, we present some generalizations of classical results from module theory, in the two-sided and graded sense. Let G be a group, F an algebraically closed field with char (F) = 0 , A a finite dimensional G-graded associative F -algebra and M a G-graded unitary A -bimodule. We proved that if A = M n (F σ [ H ]) with an elementary-canonical G-grading, where H is a finite abelian subgroup of G and σ ∈ Z 2 (H , F *) , then M being irreducible graded implies that there exists a nonzero homogeneous element w ∈ M satisfying M = B w and B w = w B . Another result we proved generalizes the last one: if G is abelian, A is simple graded and M is finitely generated, then there exist nonzero homogeneous elements w 1 , w 2 , ... , w n ∈ M such that M = A w 1 ⊕ A w 2 ⊕ ⋯ ⊕ A w n , where w i A = A w i ≠ 0 for all i = 1 , 2 , ... , n , and each A w i is irreducible. The elements wi's are associated with the irreducible characters of G. We also describe graded bimodules over graded semisimple algebras. And we finish by presenting a Pierce decomposition of the graded Jacobson radical of any finite dimensional F -algebra with a G-grading. [ABSTRACT FROM AUTHOR]

Let R[x] be the polynomial ring over a ring R with unity. A polynomial f(x) ∈ R[x] is referred to as a left annihilating content polynomial (left ACP) if there exist an element r ∈ R and a polynomial g(x) ∈ R[x] such that f(x) = rg(x) and g(x) is not a right zero-divisor polynomial in R[x]. A ring R is referred to as left EM if each polynomial f(x) ∈ R[x] is a left ACP. We observe the structure of left EM rings with various properties, and study the relationships between the one-sided EM condition and other standard ring theoretic conditions. Moreover, several extensions of EM rings are investigated, including polynomial rings, matrix rings, and Ore localizations. [ABSTRACT FROM AUTHOR]

This work is part of the overarching question of whether it is possible for the universal enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. The main result of this paper is that the universal enveloping algebra of any Krichever-Novikov algebra is not noetherian, extending a result of Sierra and Walton on the Witt (or classical Krichever-Novikov) algebra. As a subsidiary result, which may be of independent interest, we show that if h is a Lie subalgebra of g of finite codimension, then the noetherianity of U U (h) is equivalent to the noetherianity of U U (g) . The second part of the paper focuses on Lie subalgebras of W≥− 1 = Der(한[t]). In particular, we prove that certain subalgebras of W≥− 1 (denoted by L(f), where f ∈ 한[t]) have non-noetherian universal enveloping algebras, and provide a sufficient condition for a subalgebra of W≥− 1 to have a non-noetherian universal enveloping algebra. Furthermore, we make significant progress on a classification of subalgebras of W≥− 1 by showing that any infinite-dimensional subalgebra must be contained in some L(f) in a canonical way. [ABSTRACT FROM AUTHOR]

We study the growth and Gelfand-Kirillov dimension (GK-dimension) of the generalized Weyl algebra (GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and can be applied to determine the growth, GK-dimension, simplicity and cancellation properties of some GWAs. [ABSTRACT FROM AUTHOR]

The finite dual $H^{\circ}$ of an affine commutative-by-finite Hopf algebra H is studied. Such a Hopf algebra H is an extension of an affine commutative Hopf algebra A by a finite dimensional Hopf algebra $\overline{H}$. The main theorem gives natural conditions under which $H^{\circ}$ decomposes as a crossed or smash product of $\overline{H}^{\ast}$ by the finite dual $A^{\circ}$ of A. This decomposition is then further analysed using the Cartier–Gabriel–Kostant theorem to obtain component Hopf subalgebras of $H^{\circ}$ mapping onto the classical components of $A^{\circ}$. The detailed consequences for a number of families of examples are then studied. [ABSTRACT FROM AUTHOR]

We prove that if M is a C4-module (D4-module) with the restricted ascending (descending) chain condition on direct summands, then M satisfies the finite exchange iff M is clean, iff M satisfies the full exchange. This result extends an earlier one by the authors on summand-square-free (summand-dual-square-free) modules. Moreover, as an immediate application, it follows that every exchange ring with the restricted DCC on direct summands is clean. The later result is a non-trivial and simultaneous extension of two fundamental results on exchange rings, the first of which is due to Nicholson and says that Abelian exchange rings are clean, and the second is due to Camillo and Yu and says that exchange rings with no infinite sets of orthogonal idempotents are clean. [ABSTRACT FROM AUTHOR]

Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to noncommutative setting and also give new properties of amalgamated rings.

As a proper generalization of pseudo-injectivity, we introduce and investigate gs-pseudo-injective modules. A right module M over a ring R is said to be gs-pseudo-N-injective (where N is a right R-module) if every R-monomorphism from a submodule of soc (N) soc (RR) into M extends to N. A module M is said to be gs-pseudo-injective, if M is gs-pseudo-M-injective. We obtain various characterizations and properties of these modules. Some results on pseudo-injectivity are extended to gs-pseudo-injectivity. [ABSTRACT FROM AUTHOR]

Let C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring O (C) of C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work. [ABSTRACT FROM AUTHOR]

Two direct-summands A and B of a right R-module M will be called strongly-perspective (s-perspective) if for any X ⊆ M , M = A ⊕ X iff M = B ⊕ X. The module M will be called s-perspective if every two isomorphic direct-summands of M are s-perspective, and a ring R will be called s-perspective if R as a right R-module is s-perspective. In this paper, we study s-perspective rings and modules, and using ideas of Khurana and Nielsen, it follows that if M is s-perspective, then M has the finite exchange property iff M is clean, iff M has the full exchange property. In particular, if M is either summand-square-free or dual-summand-square-free with the finite exchange property, then M is clean, s-perspective and has the full exchange property. [ABSTRACT FROM AUTHOR]

In this investigation we give a module-theoretic counterpart of the well known De Morgan's laws for rings and topological spaces. We observe that the module-theoretic De Morgan's laws are related with semiprime modules and modules in which the annihilator of any fully invariant submodule is a direct summand. Also, we give a general treatment of De Morgan's laws for ordered structures (idiomatic-quantales). At the end, the manuscript goes back to the ring theoretic realm, in this case we study the non-commutative counterpart of Dedekind domains, and we describe Asano prime rings using the strong De Morgan law. [ABSTRACT FROM AUTHOR]

Let R be a Noetherian integral domain which is also an algebra over ℚ (ℚ is the field of rational numbers). Let σ be an endo-morphism of R and δ a σ-derivation of R. We recall that a ring R is a weak (σ, δ)-rigid ring if a(σ(a)+ δ(a)) ∈ N(R) if and only if a ∈ N(R) for a ∈ R (N(R) is the set of nilpotent elements of R). With this we prove that if R is a Noetherian integral domain which is also an algebra over ℚ, σ an automorphism of R and δ a σ-derivation of R such that R is a weak (σ, δ)-rigid ring, then N(R) is completely semiprime.

The Hecke-Kiselman algebra of a finite oriented graph Θ over a field K is studied. If Θ is an oriented cycle, it is shown that the algebra is semiprime and its central localization is a finite direct product of matrix algebras over the field of rational functions K(x). More generally, the radical is described in the case of PI-algebras, and it is shown that it comes from an explicitly described congruence on the underlying Hecke-Kiselman monoid. Moreover, the algebra modulo the radical is again a Hecke-Kiselman algebra and it is a finite module over its center. [ABSTRACT FROM AUTHOR]

We introduce the idea of nei-Noetherian and nei-Artinian modules and rings. We discuss some examples of nei-Noetherian (nei-Artinian) modules and study several properties of them. We characterize these modules such that an R module M is nei-Noetherian (nei-Artinian) if and only if every non essential submodule of M is iso-Noetherian (iso-Artinian) if and only if every proper closed submodule of M is iso-Noetherian (iso-Artinian). We further characterize semiprime right semihereditary nei-Noetherian rings. Finally, we discuss some other variants of ascending and descending chain conditions on the classes of non summands. [ABSTRACT FROM AUTHOR]

Generalised quantum determinantal rings are the analogue in quantum matrices of Schubert varieties. Maximal orders are the noncommutative version of integrally closed rings. In this paper, we show that generalised quantum determinantal rings are maximal orders. The cornerstone of the proof is a description of generalised quantum determinantal rings, up to a localisation, as skew polynomial extensions. [ABSTRACT FROM AUTHOR]

We prove that every A B 5 ∗ -module has an indecomposable decomposition. As an immediate consequence, every A B 5 ∗ -module M with the finite exchange is clean and has the full exchange. Moreover, in this case the module M admits an indecomposable decomposition M = ⊕ i ∈ I M i with each Mi having local endomorphism ring, and the decomposition complements direct summands. [ABSTRACT FROM AUTHOR]

Let Λ (0 0) = ( A A N B B M A B ) be a Morita ring, where the bimodule homomorphisms ϕ and ψ are zero. We study the finite presentedness, locally coherence, pure projectivity, pure injectivity, and FP-injectivity of modules over Λ(0,0). Some applications are then given. [ABSTRACT FROM AUTHOR]

We study the rings R whose injective hull E(RR) is cyclic, extending and simplifying many of the known results on the subject, and obtaining new ones. For example, we prove that if R is any ring such that E(RR) is cyclic and Dedekind-finite, then R is right self-injective. Moreover, in this case, the ring R turns out to be right co-Hopfian as well as left and right Hopfian. In particular, if E(RR) is cyclic, then R is right self-injective in the cases where R is commutative, finite-dimensional, semilocal, or strongly π-regular. We also investigate the Dedekind-finiteness of U-rings and modules, in particular those with cyclic injective hulls. [ABSTRACT FROM AUTHOR]

In this article, we continue the study of ideals of the noncommutative rings of polynomial type known as skew Poincaré-Birkhoff-Witt extensions. More exactly, we focus on the associated prime ideals of these extensions. Our purpose here is to describe how the associated primes of a ring of coefficients behave under passage to a skew PBW extension over this ring. As we show, all results established here coincide and generalize those corresponding in the commutative case and the Ore extensions. [ABSTRACT FROM AUTHOR]

In this article, we introduce S-multiplication modules which are a generalization of multiplication modules. Let M be an R-module and S ⊆ R a multiplicatively closed subset. M is said to be an S-multiplication module if for each submodule N of M there exist s ∈ S and an ideal I of R such that s N ⊆ I M ⊆ N. Besides giving many properties of S-multiplication modules, we generalize some results on multiplication modules to S-multiplication modules. Also, we study S-prime submodules in S-multiplication modules. In particular, we generalize prime avoidance lemma for multiplication modules to S -multiplication modules. Furthermore, we characterize multiplication modules in terms of S-multiplication modules. Communicated by Toma Albu [ABSTRACT FROM AUTHOR]

Let G be a group with neutral element e and let S = ⊕ g ∈ G S g be a G-graded ring. A necessary condition for S to be noetherian is that the principal component Se is noetherian. The following partial converse is well-known: If S is strongly-graded and G is a polycyclic-by-finite group, then Se being noetherian implies that S is noetherian. We will generalize the noetherianity result to the recently introduced class of epsilon-strongly graded rings. We will also provide results on the artinianity of epsilon-strongly graded rings. As our main application we obtain characterizations of noetherian and artinian Leavitt path algebras with coefficients in a general unital ring. This extends a recent characterization by Steinberg for Leavitt path algebras with coefficients in a commutative unital ring and previous characterizations by Abrams, Aranda Pino and Siles Molina for Leavitt path algebras with coefficients in a field. Secondly, we obtain characterizations of noetherian and artinian unital partial crossed products. [ABSTRACT FROM AUTHOR]

We study a new (large) class of algebras (that was introduced in Bavula in Math Comput Sci 11(3–4):253–268, 2017)—the skew category algebras. Any such an algebra C (σ) is constructed from a category C and a functor σ from the category C to the category of algebras. Criteria are given for the algebra C (σ) to be simple or left Noetherian or right Noetherian or semiprime or have 1. [ABSTRACT FROM AUTHOR]

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with M = A + B and M / A ≅ M / B. The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is dual-square-free. In this article we consider certain classes of rings R over which E (R R) is a DSF-module. For example, if R is a right hereditary ring such that E (R R) is DSF, then R is right noetherian, right distributive, every right ideal of R is two-sided, and every subfactor of RR is quasi-continuous. Also, if R is semilocal and E (R R) is a DSF-module with small radical, then R is basic, semiperfect and right self-injective. As an immediate application, if R is a right perfect ring such that E (R / J 2) is DSF as a right R-module then R is right Artinian. If in addition we assume that either E (R R) or E (R R) is a DSF-module then the ring R becomes quasi-Frobenius. [ABSTRACT FROM AUTHOR]