Almost-free E(R)-algebras and E(A,R)-modules

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.

The rings of the title are characterized. In view of earlier work on this problem, the main contribution here is the following result. Let R be a (commutative unital) ring extension of ℤ of the form ℤ[t] which is not an integral domain and which is not integral over ℤ. Then R has only finitely many (unital) subrings if and only if there exist nonzero integers a, b with at = b such that the minimal positive such a and the corresponding b satisfy (i) is integral over ℤ and (ii) there does not exist a prime number p such that ker(ϕ)⊆ p ℤ[X], where ϕ is the (unital) ring homomorphism ℤ[X] → R sending X to t. It is also proved that if T is any finite ring and n any nonzero integer, then T × ℤ[1/n] has only finitely many subrings.

Given any finitely aligned higher-rank graph Λ and any unital commutative ring R, the Kumjian–Pask algebra KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is known as the higher-rank generalization of Leavitt path algebras. After the characterization of simple Kumjian–Pask algebras by Clark and Pangalela among others, in this article we focus on the purely infinite simple ones. Briefly, we show that if KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is simple and every vertex of Λ is reached from a generalized cycle with an entrance, then KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is purely infinite. We also prove a dichotomy for simple Kumjian–Pask algebras: If each vertex of Λ is reached only from finitely many vertices and KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is simple, then KP R ⁢ ( Λ ) {\mathrm{KP}_{R}(\Lambda)} is either purely infinite or locally matritial. This result covers all unital simple Kumjian–Pask algebras.

Let R be a unital commutative ring and G an ample groupoid. Using the topology of the groupoid G , Steinberg defined an étale groupoid algebra R G . These étale groupoid algebras generalize various algebras including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for étale groupoid algebras. We characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple étale groupoid algebras, generalizing existing results for Leavitt path algebras and introducing new results for inverse semigroup algebras.

Let be the algebra of all n × n upper triangular matrices over a commutative unital ring , and let be a 2-torsion free unital -bimodule. We show that every Lie derivation is a sum of a derivation and a linear map having its range in the center of . We also consider the question of innerness of derivations from into .

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.