Applications of reduced and coreduced modules I

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 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 $P$.

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._

Symmetric polynomials and Hall's theorem

Let $R$ be a unital commutative ring and $A$ a unital $R$-algebra. We introduce the category of $E(A,R)$-modules which is a natural extension of the category of $E$-modules. The properties of $E(A,R)$-modules are studied; in particular we consider the sub

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.