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Following a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P 38 is isomorphic to the anticanonically embedded weighted projective space P (1 , 1 , 4 , 6) . [ABSTRACT FROM AUTHOR]

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let W be a classical Weyl group of rank n, with W ˜ its corresponding affine Weyl group. Our main result is that if X is a conjugacy class of involutions in W ˜ , then the commuting involution graph C (W ˜ , X) is either disconnected or has diameter at most n + 2. This bound is known to hold for types A ˜ n and C ˜ n , so the main work of this paper is to prove the theorem for types B ˜ n and D ˜ n. [ABSTRACT FROM AUTHOR]

This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”. [ABSTRACT FROM AUTHOR]

We analyze the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebraHon thenth Weyl algebraA = An(K) over a fieldKof characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebraHon an iterated Ore extension of derivation typein characteristic zero factors through a group algebra. [ABSTRACT FROM AUTHOR]

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a "non-commutative family" of central simple algebras and thus parameterizes Brauer classes over extensions of the base. [ABSTRACT FROM AUTHOR]

AbstractThis paper is concerned with computing the number of orbits under the action of the algebraic group E=E6(K) on the set of all pairs of root subgroups of itself. Furthermore, the number of orbits of the action of the group generated by a pair of certain reflections, on the 27 dimensional module A, has been computed. For the computations done in this paper, an explicit representation of E inside GL(A) has been fixed right at the onset. [ABSTRACT FROM AUTHOR]

AbstractThis paper extends the Isaacs property, originally introduced by the second author, to noncommutative fusion rings through the use of a Fourier transform. It aligns this property with a subsequent categorical equivalent for spherical fusion categories described by Etingof et al., particularly in the pseudounitary case. Within the commutative framework, the Isaacs property is situated strictly between the integrality of structure constants and the 1- Frobenius properties. Furthermore, the paper demonstrates that the Extended Haagerup fusion categories (denoted as EHi) do not possess the Isaacs property, thus providing a negative answer to questions posed by the aforementioned researchers. It also reaffirms that EH1 lacks a braiding structure. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, S ⊆ R be a multiplicative set, and M be an R-module. We say that a submodule N of M with (N : R M) ∩ S = ∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules. [ABSTRACT FROM AUTHOR]

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [ABSTRACT FROM AUTHOR]

Let be a proper closed subset in the projective n-space over a field k of characteristic zero and let C(Y) be its affine cone. Let be the homogeneous defining ideal of Y with homogeneous maximal ideal. In this paper, we find out a lower bound of de Rham depth of C(Y) in terms of. [ABSTRACT FROM AUTHOR]

In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

AbstractIn a recent paper, Baker and Oliver Lorscheid showed that Descartes’s Rule of Signs and Newton’s Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

A theorem in the article “Characterization of Monoids by Properties of Generators,” written by Kilp and Knauer, states that amalgamated coproducts of generators are generators in the categoryAct-S of right S-acts over a monoid S. In this note we give a counterexample to deny this theorem. However, we show that the results in that article based on the mentioned theorem still remain valid. Moreover, an incorrect proposition in the monographMonoids, Acts and Categoriesby Kilp et al. which states that coamalgamated product of cogenerators are cogenerators is denied by a counterexample. [ABSTRACT FROM PUBLISHER]

In [2003], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively by Beshaj, Shaska, and Shor to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the Yamauchi's formula works only under certain conditions. [ABSTRACT FROM AUTHOR]

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. [ABSTRACT FROM AUTHOR]

Let o (G) be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between o (G) and o (H) , where H is a subgroup of G. Let N be the class of finite nilpotent groups and let L (G) be the subgroup lattice of G. In this paper, we show that the set { o (G) o (H) | G ∈ N , H ∈ L (G) } is dense in [ 0 , ∞) . Other density results are outlined throughout the paper. [ABSTRACT FROM AUTHOR]

In this paper, we show that the cover ideals of theta graphs are normal square-free monomial ideals, and also satisfy the strong persistence property. Moreover, we re-prove a well-known result which is related to the cover ideals of bipartite graphs. We finally terminate this paper by stating some theorems which introduce some classes of normal monomial ideals. [ABSTRACT FROM AUTHOR]

In the present paper we study the Jordan counterparts of Rickart and Baer * -algebras, i.e., inner RJ-algebras and inner BJ-algebras and prove that a nilpotent Jordan algebra which has no square root nilpotent elements is an inner RJ-algebra. Also we explain that a nilpotent Jordan algebra that has no nilpotent elements with a square root b such that b 3 ≠ 0 is not an inner RJ-algebra if there exists a nonzero element a such that a 2 ≠ 0 . As a main result of the paper we give a description of a finite-dimensional inner RJ-algebra A , isomorphic to R + ̇ N , with a nilradical N and a finite-dimensional inner BJ-algebra with a nilradical N. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) the edge ring k [ G ] is k [ x 1 , ... , x n ] / I (G) , where n = | V | and I(G) is generated by { x i x j ; { i , j } ∈ E } . The conjecture we treat is the following. Conjecture 1. If k [ G ] has a 2-linear resolution, then the projective dimension of K [ G ] , pd (k [ G ]) , equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting k [ G ] as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds. [ABSTRACT FROM AUTHOR]

This very short correction notes a gap in an argument of an earlier paper, and also provides a theorem of similar flavor to the main result of that paper. [ABSTRACT FROM AUTHOR]

In this paper, we progress in the general study of rational functional identities in the setting of division rings. The main theorem in the paper states that the only additive maps f and g satisfying that f (x) = − x n g (x − 1) , for n = 3 or 4, are the zero maps. In addition, we provide an alternative proof to the result proved by Dar and Jing regarding the classification of additive maps f and g satisfying that f (x) = − x 2 g (x − 1) . [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

A ring idempotent e ∈ R is said to be quarter-central (or q-central for short) if e R (1 − e) R e = 0 . If all idempotents in a ring R are q-central, R is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring R is characterized by the property that [ e , R ] [ e , R ] = 0 for all idempotents e ∈ R , where [ x , y ] denotes the additive commutator x y − y x . For any nonzero ring S and any integer n ≥ 3 , we show that the ring T n (S) of n × n upper triangular matrices over S is not q-abelian. On the other hand, T 2 (S) is q-abelian iff S is abelian (in the classical sense that all idempotents are central in S ). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we give a characterization of finite non-solvable groups which act flag-regularly on maps with negative square-free Euler characteristic. We also construct some interesting families of regular maps. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, by using a combinatorial approach, we establish a new upper bound for the F-threshold cm(m) of determinantal rings generated by maximal minors. We prove that cm(m) coincides with the negative a-invariant in the case of 3×n and 4×n matrices and we conjecture such equality holds for all matrices. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we give a complete classification of symplectic structures on six-dimensional Frobeniusian solvable Lie algebras, up to symplectomorphism. We provide a scheme to classify the isomorphism classes of six-dimensional Frobeniusian solvable Lie algebras whose exact form has a Lagrangian ideal. We complete our classification by considering Frobeniusian solvable Lie algebras without Lagrangian ideal. [ABSTRACT FROM AUTHOR]

AbstractIn 2017, Igusa and Todorov gave a bijection between signed exceptional sequences and ordered partial clusters. In this paper, we show that every term in an exceptional sequence is either relatively projective or relatively injective or both and we refine this bijection to one between projectively signed exceptional sequences and ordered partial positive clusters. We also give a characterization of relatively projective/injective objects in terms of supports of the objects in the exceptional sequence. [ABSTRACT FROM AUTHOR]

Abstract–In this paper, we study additive maps f:A→A on an alternative Cayley-Dickson algebra A satisfying the product preserving property: f(x)f(y)=m whenever xy = k, where k and m are constant elements of A . We prove that f may be written as a multiple of a Jordan homomorphism. Unlike in the associative case, Jordan homomorphisms of strictly alternative division rings need not be homomorphisms or anti-homomorphisms, but we provide necessary and sufficient conditions for this to occur. We then extend these ideas to split alternative Cayley-Dickson algebras of characteristic not 2 and present some open questions when k or m is noninvertible. The final section handles the problem k=m=0 and generalizes to biadditive maps that preserve the zero product. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we consider the orthosymplectic Lie superalgebra osp(1,2) over an algebraically closed field of characteristic p>2 . We give a sufficient and necessary condition for a map from the cartesian product of a finite-dimensional Lie superalgebra with itself to its nontrivial and simple modules to be a symmetric super-biderivation. Finally, we determine all super-biderivations from osp(1,2) to its any finite-dimensional simple module. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we partially answer a question of Wujie Shi about the width of the order of the elements of a finite group. [ABSTRACT FROM AUTHOR]

AbstractLet G be a finite group, and let p be a prime. In this paper, we obtain some conditions for G to be p-nilpotent or p-closed in terms of strongly monolithic characters. [ABSTRACT FROM AUTHOR]

AbstractThe aim of this paper is to investigate the influence of intrinsic properties of maximal subgroups on Coleman automorphisms of finite groups G. Let M be a maximal subgroup of G. It is proved that if M is simple then OutCol(G)=1 . It is also proved that either OutCol(G) is trivial or OutCol(G) is an abelian p-group for some prime p provided that M has a unique nontrivial proper normal subgroup. As an application, it is obtained that under some additional conditions the normalizer property holds for such groups. [ABSTRACT FROM AUTHOR]

This paper centers around one of several generalizations of primary ideals. Which is an intermediate class between S-primary ideals and quasi S-primary ideals. Let R be a commutative ring with identity and S be a multiplicative closed subset of R. A proper ideal I of R disjoint from S is called strongly quasi S-primary if there exists an s ∈ S such that whenever x , y ∈ R and xy ∈ I , then either s x 2 ∈ I or sy ∈ I . Many basic properties of strongly quasi S-primary ideals are given, and examples are presented to distinguish the last concept from other classical ideals. Moreover, forms of strongly quasi S-primary ideals in polynomial rings, power series rings and idealization of a module are investigated. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension r a (M) , where r a (M) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set Coass R H r a (M) a (M) finite, we show that r a (M) = inf { r a R p ( Hom R (R p , M)) | p ∈ Spec R } . And the class of all modules N such that Coass R N is finite is studied. [ABSTRACT FROM AUTHOR]

In this paper we study certain combinatorial attributes of Ind -schemes of polynomial automorphisms in positive characteristic. In particular, we prove that over an algebraically closed field K of positive characteristic ≠ 2 every automorphism of the group of origin-preserving automorphisms of the polynomial algebra K [ x 1 , ... , x n ] ( n ≥ 3 ), which fixes every diagonal matrix, preserves, up to composition with a linear inner automorphism, every tame automorphism. [ABSTRACT FROM AUTHOR]

Let V be a vector space and U a fixed subspace of V. We denote the semigroup of all linear transformations on V under composition of functions by L(V). In this paper, we study the semigroup of all linear transformations on V whose restrictions belong to the general linear group GL(U), denoted by L GL (U) (V) . More precisely, we consider the subsemigroup L GL (U) (V) = { α ∈ L (V) : α | U ∈ GL (U) } of L(V). In this work, Green's relations and ideals of this semigroup are described. Then we also determine the minimal ideal and the set of all minimal idempotents of it. Moreover, we establish an isomorphism theorem when V is a finite dimensional vector space over a finite field. Finally, we find its generating set. [ABSTRACT FROM AUTHOR]

The paper studies the interplay between MV-algebras, Bézout domains, and Abelian l -groups, which was initiated by Y. C. Yang in 2006. For each MV-algebra M = Γ (G , u) , we delve into some properties of a subset of a Bézout domain corresponding to M. We study Boolean elements and ideals within the MV-algebra, focusing on their correspondence to subsets of Bézout domains. This investigation aids us in classifying Bézout domains about directly indecomposable MV-algebras or subdirect products of MV-algebras. We establish that each MV-algebra associated with a Noetherian Bézout domain is finite. In conclusion, from a ring-theoretical perspective, we present characteristics of perfect MV-algebras, (H , 1) -perfect MV-algebras, hyperarchimedean MV-algebras, and complete MV-algebras. [ABSTRACT FROM AUTHOR]

The aim of this paper is to examine the structure of genetic algebras associated with permuted Lotka-Volterra operators on a finite-dimensional simplex. Furthermore, three dimensional permuted Lotka-Voltarra algebras are considered and the description of their derivations are given as well. [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π -property in G if there exists a chief series Γ G : 1 = G 0 < G 1 < · · · < G n = G of G such that for every G-chief factor G i / G i − 1 (1 ≤ i ≤ n) of Γ G , | G / G i − 1 : N G / G i − 1 (H G i − 1 / G i − 1 ∩ G i / G i − 1) | is a π (H G i − 1 / G i − 1 ∩ G i / G i − 1) -number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial Π -property on the structure of a finite group. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of curved O -operator systems as a generalization of T. Brzeziński's (curved) Rota-Baxter systems, and then investigate their relations with O -operator systems, (tri)dendriform systems, pre-Lie algebras, associative Yang-Baxter pairs and quasitriangular covariant bialgebras. [ABSTRACT FROM AUTHOR]

Let F be a class of finite groups. A group G is called a minimal non- F -group or simply an F -critical group, if G is not in F but all proper subgroups of G are in F . Let Sch(G) denotes the set of all Schmidt subgroups of a group G (i.e. N -critical subgroups of G, where N is the class of all nilpotent groups), and Sch U = { G | Sch (G) ⊆ U } , where U is the class of all supersoluble groups. In the paper, we investigate the new properties of the class Sch U . In particular, we prove that Sch U is a subgroup-closed saturated Fitting formation with Shemetkov property, i.e. every Sch U -critical group is a Schmidt group. In addition, we show that Sch U is locally defined by the formation function f such that f (p) = S { p } ∪ π (p − 1) for every prime p. Besides, we describe all Sch U -critical groups and all minimal non-supersoluble groups that belong to Sch U . [ABSTRACT FROM AUTHOR]

For a simple graph G with edge ideal I(G), we study the N -graded Betti numbers in the linear strand of the minimal free resolution of I (Γ (Z n)) , where Γ (Z n) is the zero divisor graph of the ring Z n . We present sharp bounds for the Betti numbers of Γ (Z n) and characterize the graphs attaining these bounds, thereby establishing the correct equality case for one of the results of the earlier published paper (Theorem 4.5, S. Pirzada and S. Ahmad, On the linear strand of edge ideals of some zero divisor graphs, Commun. Algebra 51(2) (2023) 620–632). Also, we present homological invariants of the edge rings of Γ (Z n) for n = p 2 q and pqr, with primes p < q < r. [ABSTRACT FROM AUTHOR]