Narrow Results

We correct [3], Proposition 3.2] by showing that the class of quasi-Noetherian Lie algebras s is not $\mathrm{E}$-closed. We repair the resulting gap in Example 5.1 of that paper by proving that any simple-by-soluble Lie algebra is quasi-Noetherian. More generally, any Noetherian-by-quasi-Noetherian Lie algebra is quasi-Noetherian. We also prove that any quasi-Noetherian-by-soluble Lie algebra is quasi-Noetherian, and prove $\mathrm{E}$-closure for analogues of the quasi-Noetherian property for modules over associative rings and modules over Lie algebras. Using a wreath product for Lie algebras we prove that the quasi-Noetherian property is not inherited by ideals. Indeed, the derived algebra of a quasi-Noetherian Lie algebra need not be quasi-Noetherian. An analogous example is constructed for quasi-Artinian Lie algebras, which also shows that the Artinian and quasi-Artinian properties are not inherited by ideals of finite codimension.

One of the central themes in the research of function rings over topological spaces is build new rings, build new spaces. In this paper, we continue the work of Knox on the ring of density continuous functions, denoted by $C(X,{ℝ}_d)$. We infer that $C(X,{ℝ}_d)$ is self-injective (${\aleph }_0$-self-injective) if the family of zero-sets, denoted by $Z_d[X]$ is closed under arbitrary (countable) intersections. We provide a counterexample of the converse statement on self-injectivity, and in that process, we answer a question raised in [H. Azadi, M. Henriksen and E. Momtahan, Some properties of algebra of real valued measurable functions, Acta Math. Hunger124(1–2) (2009) 15–23]. We have established a condition on $X$ for which $C(X,{ℝ}_d)$ is Artinian and Noetherian. Unlike $C(X)$, the zero-divisor graph of $C(X,{ℝ}_d)$ is seen to be neither triangulated nor hyper-triangulated. We prove that space $X$ is density compact if and only if it is density pseudocompact and density real compact.

We consider algebras over a field with generators x₁,x₂,…,x_n subject to square-free relations x_ix_j=x_kx_l in which every product x_px_q, p≠q, appears in one of the relations. The work of Gateva-Ivanova and Van den Bergh, motivated in particular by the study of set theoretic solutions of the Yang–Baxter equation, provided an important class of such algebras. In this special case the presentation satisfies the so-called cyclic condition that became an essential combinatorial tool in proving that these algebras share many strong ring theoretic properties of polynomial algebras in commuting variables. In this paper we describe the structure of algebras on four generators satisfying the cyclic condition. The emphasis is on some new unexpected features, not present in the motivating special classes.

Symmetric radicals over a fully semiprimary Noetherian ring R are characterized in terms of stability on bimodules and link closure of special classes of prime ideals. The notion of subdirect irreduciblity with respect to a torsion radical is introduced and is shown to be invariant under internal bonds between prime ideals. An analog of the Jacobson radical is produced which is properly larger than the Jacobson radical, yet satisfies the conclusion of Jacobson's conjecture for right fully semiprimary Noetherian rings.

Over an arbitrary ring R, a symmetric radical is shown to be strongly normalizing. For a fully semiprimary Noetherian ring R, a symmetric radical is normalizing if and only if the class of torsion factor rings of R is closed under ring isomorphisms. In case S is a strongly normalizing or normalizing extension ring of R, a symmetric radical for S is constructed as an extension of a symmetric radical for R. Applications address questions concerning the behavior of Krull dimension and linked prime ideals of S.

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if for every positive integer n and distinct maximal submodules L, L_i (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes , such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

We study direct sum decompositions of modules satisfying the descending chain condition on direct summands. We call modules satisfying this condition Krull–Schmidt artinian. We prove that all direct sum decompositions of Krull–Schmidt artinian modules refine into finite indecomposable direct sum decompositions and we prove that this condition is strictly stronger than the condition of a module admitting finite indecomposable direct sum decompositions. We also study the problem of existence and uniqueness of direct sum decompositions of Krull–Schmidt artinian modules in terms of given classes of modules. We present also brief studies of direct sum decompositions of modules with deviation on direct summands and of modules with finite Krull–Schmidt length.

In this note, we generalize the results of [Submaximal integral domains, Taiwanese J. Math. 17(4) (2013) 1395–1412; Which fields have no maximal subrings? Rend. Sem. Mat. Univ. Padova 126 (2011) 213–228; On the existence of maximal subrings in commutative artinian rings, J. Algebra Appl. 9(5) (2010) 771–778; On maximal subrings of commutative rings, Algebra Colloq. 19(Spec 1) (2012) 1125–1138] for the existence of maximal subrings in a commutative noetherian ring. First, we show that for determining when an infinite noetherian ring R has a maximal subring, it suffices to assume that R is an integral domain with |R/I| < |R| for each nonzero ideal I of R. We determine when the latter integral domains have maximal subrings. In particular, we show that every uncountable noetherian ring has a maximal subring.

In this note, we classify the non-Noetherian generalized Heisenberg algebras ${\mathscr{H}}(f)$ introduced in [R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl.475 (2015) 276–291]. In case deg $f$ > 1, we determine all locally finite and also all locally nilpotent derivations of ${\mathscr{H}}(f)$ and describe the automorphism group of these algebras.

The paper studies the transfer of the property of being right G-semilocal and right N-semilocal from a ring $R$ to its extensions and vice versa. We will focus on two extensions, namely finite normalizing extension $S$ of $R$ and the ring of polynomials $R[x]$.

A Lie algebra (over any field and of any dimension) is Noetherian if it satisfies the maximal condition on ideals. We introduce a new and more general class of quasi-Noetherian Lie algebras that possess several of the main properties of Noetherian Lie algebras. This class is shown to be closed under quotients and extensions. We obtain conditions under which a quasi-Noetherian Lie algebra is Noetherian. Next, we consider various questions about locally nilpotent and soluble radicals of quasi-Noetherian Lie algebras. We show that there exists a semisimple quasi-Noetherian Lie algebra that is not Noetherian. Finally, we consider some analogous results for groups and prove that a quasi-Noetherian group is countably recognizable.

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter $q$ and two polynomials $f,g$. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties.

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (respectively, t-ideals $I_t(D)$) of an integral domain $D\ne qf(D)$ and let P be a valid property of ideals of D. We say S(D) meets P (denoted $S(D)◃P)$ if each $s\in S(D)$ is contained in an ideal satisfying P. If S(D) $◃P,$$dim(D)$ cannot be controlled. When $R=D[X],$$I(D)$$◃P$ does not imply I(R) $◃P$ while $I_t(D)$$◃P$ implies $I_t(R)$$◃P$ usually. We say S(D) meets P with a twist $($written $S(D){◃}^{t}P)$ if each $s\in S(D)$ is such that, for some $n\in N,$$s^n$ is contained in an ideal satisfying P and study $S(D){◃}^{t}P,$ as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.

The SFT (for strong finite type) condition was introduced by [J. T. Arnold, Krull dimension in power series rings, Trans. Amer. Math. Soc. 177 (1973) 299–304] in the context of studying the condition for formal power series rings to have finite Krull dimension. In the context of commutative rings, the SFT property is a near-Noetherian property that is necessary for a ring of formal power series to have finite Krull dimension behavior. Many others have studied this condition in the context of the dimension of formal power series rings. In this paper, we explore a specialization (and in some sense a more natural) variant of the SFT property that we dub as the VSFT (for very strong finite type) property. As is true of the SFT property, the VSFT property is a property of an ideal that may be extended to a global property of a commutative ring with identity. Any ideal (respectively, ring) that has the VSFT property has the SFT property. In this paper, we explore the interplay of the SFT property and the VSFT property.

We study a certain type of prime Noetherian idealiser ring R of injective dimension 1, and prove for instance that the idempotent ideals of R are projective and that every non-zero projective ideal of R is uniquely of the form UE for some invertible ideal U and idempotent ideal E of R. Formulae are given for the number of idempotent ideals of R and the number of orders which contain R.

This is a modified manuscript of my talk entitled "Open Questions on Fully Prime Rings" presented at the 6th CJK-ICRT. A brief survey of results on fully prime and related rings is given and it is followed by a few results and problems that may be helpful to investigate the main conjecture: under the assumption that the ring is right and left Noetherian, a fully prime ring is right primitive.