Lie derivations on triangular matrices

UDC 512.5 Let be a commutative ring with unity and be a unital algebra over (or field ).An -linear map is called a Lie derivation on if holds for all For scalar an additive map is called an additive -Lie derivation on if where holds for all In the present paper, under certain assumptions on it is shown that every Lie derivation (resp., additive -Lie derivation) on is of standard form, i.e., where is an additive derivation on and is a mapping vanishing at with in Moreover, we also characterize the additive -Lie derivation for by its action at zero product in a unital algebra over

In this article, basic ideals in a Leavitt path algebra over a com- mutative unital ring are studied. It is shown that for a nite acyclic graph E and a commutative unital ring R, the Leavitt path algebra LR(E) is a direct sum of minimal basic ideals and that for a commutative ring R and a graph E satisfying Condition (L), the Leavitt path algebra LR(E) has no non-zero nilpotent basic ideals. Uniqueness theorems for Leavitt path algebras over commutative unital rings are also discussed.

Let S be a unital ring in which 2 is invertible, and let be the quaternion ring over S. In this paper, we describe the Lie derivations and generalized Lie derivations of R, we show that if S is commutative or semiprime, then every Lie derivation (resp. generalized Lie derivation) of R decomposes into the sum of a derivation (resp. generalized derivation) and an additive central map that vanishes on all commutators of R.

In this paper, we study the stability of specific local functional inequalities within quasi-[Formula: see text]-Banach spaces, focusing on mappings that satisfy modular concavity and bounded perturbations. The analysis establishes the existence and uniqueness of [Formula: see text]-linear mappings that approximate these functions. Furthermore, we apply these results to investigate the stability of Lie homomorphisms and Lie derivations in quasi-[Formula: see text]-Banach spaces.

Let A be a differential graded (DG) algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of graded A-modules to the category of DG A-modules. Specifically, we exhibit a special class of semifree DG A-modules which can always be expressed as the totaling of some complex of graded free A-modules. As a corollary, we also provide results concerning the image of the totaling functor when A is a polynomial ring over a field.

Let R be a commutative unital ring and N be a submodule of an R-module M. We show that: 1) the semiprime radical is an invariant on submodules generated by the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, ⟨EM (0)⟩ is an idempotent radical leading to a torsion theory whose torsion class has nil R-modules and the torsion-free class has reduced R-modules; and, 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module.

Leavitt path algebras of labelled graphs

Given any edge-colored graph G and any commutative unital ring R, we construct a generalized Leavitt path algebra LR(G).We show that LR(G) is a certain free product of LR(Gi), where Gis are 1-colored subgraphs of G. We also show that LR(G) may be written as a free product of simpler algebras. In the end, we define a natural \({\mathbb{Z}}\) -grading for LR(G) and give four necessary conditions for simplicity of LR(G).

Let $P$ be a partially ordered set, $R$ a commutative unital ring and $FI(P,R)$ the finitary incidence algebra of $P$ over $R$. We prove that each $R$-linear higher derivation of $FI(P,R)$ decomposes into the product of an inner higher derivation of $FI(P,R)$ and the higher derivation of $FI(P,R)$ induced by a higher transitive map on the set of segments of $P$.

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.

Let R be a commutative unital ring and n ∈ ℕ, n ≥ 2. A system σ = (σij), 1 ≤ i, j ≤ n, of additive subgroups σij of R is called a net or carpet over R of order n if σirσrj ⊆ σij for all i, r, and j. A net without diagonal is called an elementary net or elementary carpet. Let n ≥ 3. Consider a matrix ω = (ωij) of additive subgroups ωij of R, defined for i ≠ j as follows: $$ {\omega}_{ij}=\sum \limits_{k=1}^n{\sigma}_{ik}{\sigma}_{kj} $$ , k ≠ i, j. The set ω = (ωij) of elementary subgroups ωij of R is an elementary net ω and is called an elementary derived net. The diagonal of the derived net ω is defined by the formula $$ {\omega}_{ii}=\sum \limits_{k\ne s}{\sigma}_{ik}{\sigma}_{ks}{\sigma}_{si} $$ , 1 ≤ i ≤ n, where the sum is taken over all 1 ≤ k ≠ s ≤ n. It is proved that an elementary net σ induces the derived net ω = (ωij) and the net Ω = (Ωij) associated with the elementary group E(σ), where ω ⊆ σ ⊆ Ω, ωirΩrj ⊆ ωij and Ωirωrj ⊆ ωij (1 ≤ i, r, j ≤ n). In particular, the matrix ring M(ω) is a two-sided ideal of the ring M(Ω). For the nets of order n = 3, we establish a more precise result.

Let R be a commutative unital ring and E a unital R-module. Then the canonical injective ring homomorphism from R into the idealization R(+) E is a minimal ring homomorphism if and only if E is a simple R-module. For E nonzero, R(+)E is not (R-algebra isomorphic to) an overring of R. If E 1 and E 2 are nonisomorphic simple R-modules, then R(+) E 1 and R(+) E 2 give minimal ring extensions of R which are not isomorphic as R-algebras. The ring of dual numbers over R is a minimal ring extension of R ⇔ R × R is a minimal ring extension of R ⇔ R is a field.

Considering prime Leavitt path algebras L K ⁢ ( E ) {L_{K}(E)} , with E being an arbitrary graph with at least two vertices, and K being any field, we construct a class of maximal commutative subalgebras of L K ⁢ ( E ) {L_{K}(E)} such that, for every algebra A from this class, A has zero intersection with the commutative core ℳ K ⁢ ( E ) {\mathcal{M}_{K}(E)} of L K ⁢ ( E ) {L_{K}(E)} defined and studied in [C. Gil Canto and A. Nasr-Isfahani, The commutative core of a Leavitt path algebra, J. Algebra 511 2018, 227–248]. We also give a new proof of the maximality, as a commutative subalgebra, of the commutative core ℳ R ⁢ ( E ) {\mathcal{M}_{R}(E)} of an arbitrary Leavitt path algebra L R ⁢ ( E ) {L_{R}(E)} , where E is an arbitrary graph and R is a commutative unital ring.

Let R be a commutative unitary ring and I an ideal of R. We prove that is isonoetherian if and only if R is isonoetherian, I is idempotent and each ideal of R contained in I is finitely generated. We prove that satisfies ACC d on ideals if and only if R is Noetherian. We deduce that the ring satisfies ACC d on ideals if and only if R satisfies ACC d on ideals, I is idempotent and each ideal of R contained in I is finitely generated. We study polynomial rings satisfying ACC d on ideals. We give an example of a ring satisfying ACC d on ideals but is not isonoetherian.

Let S be a cancellative monoid with quotient group of torsion-free rank a.We show that the monoid ring R[S] is a Hilbert ring if and only if the polynomial ring R[{ X, },s/] is a Hilbert ring, where |/| = a.Assume that R is a commutative unitary ring and G is an abelian group.The first research problem listed in [K, Chapter 7] is that of determining equivalent conditions in order that the group ring R[G] should be a Hilbert ring.Matsuda has considered this question in [Ml, 6].His results show that for G finitely generated, R[G] and R are simultaneously Hilbert rings; if G is not finitely generated, then R[G] a Hilbert ring implies that R is a Hilbert ring, but the converse fails.In [M2, 5] Matsuda considers briefly the corresponding Hilbert-ring-characterization problem for a monoid ring R[S], where S is torsion-free and cancellative.Let a be the torsion-free rank of G and let {Xj)lsl be a set of indeterminates over R of cardinality a.In Corollary 1 we show that R[G] and P[{^,},e/] are simultaneously Hilbert rings.While the problem of determining equivalent conditions for P[{Ar,}/e/] to be a Hilbert ring has not been completely resolved, it has been worked on extensively [Kr, Go, L, Gl, H], and a significant body of positive results exists concerning this problem.The proof of Theorem 1 suggests the following conjecture: if S is a cancellative commutative monoid with quotient group G, then R[S] is a Hilbert ring if and only if R[G] is a Hilbert ring; this result is established in Theorem 2.All monoids considered are assumed to be commutative, and rings are assumed to be commutative and unitary.The statement of Theorem 1 uses the following terminology.For a cardinal number a, an extension ring T of R is said to be a-generated over R if T is generated over R by a set of cardinality at most a.Theorem 1. Assume that X = {XA is a set of indeterminates of cardinality a over the ring R. Denote by Za the direct sum of a copies of the additive group Z of integers.The following conditions are equivalent.(1) R[X] is not a Hilbert ring.(2) There exists a prime ideal P of R such that R/P admits an a-generated extension ring that is a G-domain, but not afield.(3) The group ring R[Za] is not a Hilbert ring._