Search

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

A proper ideal I of a commutative ring R is said to be a strongly quasi-primary ideal if, whenever a , b ∈ R with a b ∈ I , then a 2 ∈ I or b ∈ I (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of R , called the class of quasi- n -ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of R is a maximal or a minimal ideal of R. Many examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the derivation module of the ring of invariants of k [ X , Y ] under the linear action of dihedral groups D n , q < G L (2 , k) mentioned in a paper by Riemenschneider [Die Invarianten der endlichen Untergruppen von G L (2 , ℂ) , Math. Zeitsch. 153 (1977) 37–50]. We obtained an explicit generating set for the derivation module of R = k [ X , Y ] D n , q . We show that μ ( Der k (R)) ≤ O (D n , q). [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

The rings considered in this paper are commutative with identity. If R is a subring of a ring T , then we assume that R contains the identity element of T. Let S be a multiplicatively closed subset (m.c. subset) of a ring R. An increasing sequence of ideals I 1 ⊆ I 2 ⊆ I 3 ⊆ ⋯ of R is said to be S -stationary if there exist s ∈ S and k ∈ ℕ such that s I j ⊆ I k for all j ≥ k. This paper is motivated by the research work [A. Hamed and H. Kim, On integral domains in which every ascending chain on principal ideals is S -stationary, Bull. Korean Math. Soc. 57(5) (2020) 1215–1229]. Let S be a m.c. subset of an integral domain D. We say that D satisfies S -ACCP if every increasing sequence of principal ideals of D is S -stationary. Let D be a subring of an integral domain T and let S be a m.c. subset of D. We say that (D , T) is an S -ACCP pair if A satisfies S -ACCP for every subring A of T with D ⊆ A. The aim of this paper is to provide some pairs of domains D ⊆ T such that (D , T) is an S -ACCP pair, where S is a m.c. subset of D. [ABSTRACT FROM AUTHOR]

In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479], Rees gave a characterization for the normal joint reduction number zero of two 픪-primary ideals in an analytically unramified Cohen–Macaulay local ring of dimension two. Rees’ result is a generalization of Zariski’s product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees’ theorem for the ordinary powers of 픪-primary ideals I and J in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules Mr,s1(ak,bk) for a joint reduction (a,b) of I and J. Under the additional assumption that the associated graded rings of I and J have positive depth, we obtain a characterization of the joint reduction number zero of I and J in terms of the vanishing of the module M0,01(a,b), as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of Mr,s1(a,b) for any r,s ≥ 0. This gives a characterization for a vector (r,s) to be in the joint reduction lattice of I and J. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector (r,s) to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra 218(5) (2014) 904–918] for ordinary powers of ideals. [ABSTRACT FROM AUTHOR]

This paper defines the Euler–Poincaré characteristic of joint reductions of ideals which concerns the maximal terms in the Hilbert polynomial; characterizes the positivity of mixed multiplicities in terms of minimal joint reductions; proves the additivity and other elementary properties for mixed multiplicities. The results of the paper together with the results of [Thanh and Viet, Mixed multiplicities of maximal degrees, J. Korean Math. Soc.55(3) (2018) 605–622] seem to show a natural and nice picture of mixed multiplicities of maximal degrees. [ABSTRACT FROM AUTHOR]

The well-known notion of an extending module is closely linked to that of a Baer module. A right R -module M is called extending if every submodule of M is essential in a direct summand. On the other hand, a right R -module M is called Baer if for all N ≤ M , l S (N) ≤ ⊕ S S where S = End R (M). In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.  21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: "a module M is -nonsingular extending if and only if M is -cononsingular Baer". M R is called -nonsingular if ∀ φ ∈ S such that Ker φ ≤ e M , φ = 0. Moreover, M R is called -cononsingular if for any N ≤ M with φ N ≠ 0 for all 0 ≠ φ ∈ S , implies N ≤ e M. In view of this result, every Baer module which happens to be -cononsingular will automatically become an extending module. In this paper, our main focus is the study of -cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of -cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of -cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman's result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a condition on a p-subgroup of a finite group that is more general than the concept of ICΦ-subgroups, and obtain a sufficient and necessary condition for p-nilpotence of a finite group. We not only simplify the proofs of some main theorems of Gao and Li [Acta Math. Hungar. 163(1) (2021) 29–36], Kaspczyk [Acta Math. Hungar. 165(2) (2021) 355–359], and Kaspczyk [J. Algebra Appl. 23(1) (2024) 2450016], but also generalize these results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the class S - ℱ -ML of S -Mittag-Leffler modules with respect to all flat modules. We show that a ring R is S -coherent if and only if every ideal is in S - ℱ -ML, if and only if S - ℱ -ML is closed under submodules. As an application, we obtain the S -version of Chase Theorem: a ring R is S -coherent if and only if any direct product of copies of R is S -flat, if and only if any direct product of flat R -modules is S -flat. Consequently, we provide an answer to the open question proposed by Bennis and El Hajoui [On S -coherence, J. Korean Math. Soc. 55(6) (2018) 1499–1512]. [ABSTRACT FROM AUTHOR]

Let x be an element of a finite group G, and p a prime factor of the order of G. It is clear that there always exists a unique minimal subnormal subgroup containing x, say G x . We call the conjugacy class of x in G x the sub-class of x in G, see [G. Qian and Y. Yang, On sub-class sizes of finite groups, J. Aust. Math. Soc. (2020) 402–411]. In this paper, assume that G (= A B) is the product of the subgroups A and B, we investigate the solvability, p-nilpotence and supersolvability of the group G under the condition that the sub-class sizes of prime power order elements in A ∪ B are 2 2 free, p 2 free and square free, respectively, so that some known results relevant to conjugacy class sizes are generalized. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of commutative rings which is a generalization of ZD -rings and rings with Noetherian spectrum. A ring R is called strongly J -Noetherian whenever the ring R r is J -Noetherian for every non-nilpotent r ∈ R. We give some characterizations for strongly J -Noetherian rings and, among the other results, we show that if R [ [ x ] ] is strongly J -Noetherian, then R has Noetherian spectrum, which is a generalization of Theorem 2 in Gilmer and Heinzer [The Laskerian property, power series rings, and Noetherian spectra, Proc. Amer. Math. Soc.79 (1980) 13–16]. [ABSTRACT FROM AUTHOR]

It is well known that injective objects play a fundamental role in many branches of mathematics. The question whether a given category has enough injective objects has been investigated for many categories. Also, quasi-injective modules and acts have been studied by many categorists. In this paper, we study quasi-injectivity in the category of actions of an ordered monoid on ordered sets (P o s - S) with respect to embeddings. Also, we give the relation between injectivity, quasi-injectivity (with respect to embeddings), and poset completeness in the category P o s - S and some of its important subcategories. [ABSTRACT FROM AUTHOR]

Injectivity is one of the useful notions in algebra, as well as in many other branches of mathematics, and the study of injectivity with respect to different classes of monomorphisms is crucial in many categories. Also, essentiality is an important notion closely related to injectivity. Down closed monomorphisms and injectivity with respect to these monomorphisms, so-called dc-injectivity, were first introduced and studied by the authors for S -posets, posets with an action of a pomonoid S on them. They gave a criterion for dc-injectivity and studied such injectivity for S itself, and for its poideals. In this paper, we give results about dc-injectivity of S -posets, also we find some homological characterization of pomonoids and pogroups by dc-injectivity. In particular, we give a characterization of pomonoids over which dc-injectivity is equivalent to having a zero top element. Also, introducing the notion of T -injectivity for S -posets, where S = T ∪ ̇ { 1 } and 1 is externally adjoined to the posemigroup T , we find some classes of pomonoids such that for S -posets over them the Baer Criterion holds. Further, several kinds of essentiality of down closed monomorphisms of S -posets, and their relations with each other and with dc-injectivity is studied. It is proved that although these essential extensions are not necessarily equivalent, they behave almost equivalently with respect to dc-injectivity. Finally, we give an explicit description of dc-injective hulls of S -posets for some classes of pomonoids S. [ABSTRACT FROM AUTHOR]

Crawley-Boevey introduced in [Poisson structure on moduli spaces of representations, J. Algebra 325 (2011) 205–215.], the notion of H 0 -Poisson structure for associative algebras, which is the weakest condition that induces a Poisson structure on the moduli spaces of their representations. In this paper, by using a result of Armenta and Keller in [Derived invariance of the Tamarkin-Tsygan calculus of an algebra, C. R. Math. Acad. Sci. Paris 357(3) (2019) 236–240.], we show that an H 0 -Poisson structure is preserved under derived Morita equivalence. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings. [ABSTRACT FROM AUTHOR]

We show a necessary condition for Klein four symmetric pairs (G , G Γ) satisfying the condition (D.D.); that is, there exists at least one infinite-dimensional simple (, K) -module that is discretely decomposable as a ( Γ , K Γ) -module. This work is a continuation of [A criterion for discrete branching laws for Klein four symmetric pairs and its application to E 6 (− 1 4) , Int. J. Math. 31(6) (2020) 2050049]. Moreover, we define associated Klein four symmetric pairs, and we may use these tools to compute that a class of Klein four symmetric pairs do not satisfy the condition (D.D.); for example, ( 6 (− 1 4) , (4 , 2) ⊕ (2)). [ABSTRACT FROM AUTHOR]

Let τ be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China10 (2015) 91–110]. First they are used to obtain Chase's Theorem for τ -coherent rings. In particular, we obtain the w -version of Chase's Theorem, where w is the classical star operation in ideal theory. In the second half, we apply they to characterize τ -flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)]. [ABSTRACT FROM AUTHOR]

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra H , which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of H -modules is equivalent to the category H H 𝒬 𝒴 𝒟 of quasi-Yetter–Drinfeld modules. Next, we prove that H H 𝒬 𝒴 𝒟 is equivalent to the category of generalized Hopf bimodules. Finally, we show that H H 𝒬 𝒴 𝒟 is also equivalent to the category of quasi-coactions over some Majid's braided group if H is quasi-triangular. [ABSTRACT FROM AUTHOR]

In this paper, we give a recursive formula for the interesting PBW basis E A of composition subalgebras ℭ ▵ (n) of Ringel–Hall algebras ℌ ▵ (n) of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of U v + ( 𝔰 𝔩 ̂ n) from the monomial bases m (A) following [Multiplication formulas and canonical basis for quantum affine, 𝔤 𝔩 n , Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of U v + ( 𝔰 𝔩 ̂ 2) associated with modules of Loewy length ≤ 3. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras. [ABSTRACT FROM AUTHOR]

Let R ⊂ A be factorial domains containing ℚ. In this paper, we give a criterion, in terms of locally nilpotent derivations, for A to be R -isomorphic to R [ v , w ] / (c w − h (v)) , where c ∈ R is nonzero and h (v) ∈ R [ v ]. As a consequence, we retrieve a recent result due to Masuda [Families of hypersurfaces with noncancellation property, Proc. Amer. Math. Soc.145(4) (2017) 1439–1452] characterizing Danielewski hypersurfaces whose coordinate ring is factorial. We also apply our criterion to the study of triangularizable locally nilpotent R -derivations of the polynomial ring in two variables over R. [ABSTRACT FROM AUTHOR]

The aim of this work is to prove a technical result that had been stated by Boyallian et al. [Classification of finite irreducible modules over the Lie conformal superalgebra C K 6 , Comm. Math. Phys. 317 (2013) 503–546], on the degree of singular vectors of finite Verma modules over the exceptional Lie superalgebra E (1 , 6) that is isomorphic to the annihilation superalgebra associated with the conformal superalgebra C K 6 . [ABSTRACT FROM AUTHOR]

The modified quantum algebra associated to a quantum algebra was introduced by Lusztig. has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of for type . [ABSTRACT FROM AUTHOR]

The concept of internally cancellable rings has been extensively studied in the literature. This paper seeks to continue the study of these rings and find some new characterizations. It is proved that is 'IC', if and only if for each regular element , and idempotent element with , there exists such that is a unit (alternatively, unit-regular element) in and . In case the ring has the summand sum property, we indicate that is IC, if and only if for each regular element , and element with , there exists an idempotent , such that is a unit in and . [ABSTRACT FROM AUTHOR]

In this paper, we generalize the main results of Guo and Isaacs, Archiv der Mathematik 105 (2015) 215-222. We also obtain some criteria for -nilpotency of a finite group. [ABSTRACT FROM AUTHOR]

Let be an integral domain with quotient field . Call an almost GCD domain (AGCD domain) if for all , there is an integer such that is principal. We say that is a locally AGCD domain (resp., strongly locally AGCD domain) if is an AGCD domain for all maximal ideals of (resp., for all , there is an integer such that is locally principal). In this paper, we study some ring-theoretic properties of locally and strongly locally AGCD domains. Let be an indeterminate over . We use the ring for an integer to give some examples of locally and strongly locally AGCD domains. [ABSTRACT FROM AUTHOR]

In this paper, we prove an Artin type theorem for semigroups. Namely, we consider the concepts of hyperalternative and hyperassociative semigroups and prove that every two elements in any hyperalternative semigroup generate a hyperassociative subsemigroup. As a consequence, we characterize all hyperalternative semigroups, and prove that these semigroups form a variety of semigroups with four identities. [ABSTRACT FROM AUTHOR]

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61-68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of , the complex group algebra of . One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if is an odd prime number, then is uniquely determined by the structure of its complex group algebra. [ABSTRACT FROM AUTHOR]

We generalize Ye's Theorem which states that the group ring is a semi-clean ring [Y. Ye, Semiclean rings, Comm. Algebra 31 (2003) 5609-5625]. The proof provided here is more efficient; it is less algorithmic but has the feature that the following statement is evident: for distinct primes , the group ring is feebly clean if and only if the order of modulo is at least . [ABSTRACT FROM AUTHOR]

Since topoi were introduced, there have been efforts putting mathematics into the context of topoi. Amongst known topoi, the topoi of sheaves or presheaves over a small category are of special interest. We have here as the base topos that of sheaves over a monoid M as a one object category. By means of closure operators we then obtain categories of sheaves related to the right ideals of M. These categories have already been studied but we give these categories a more thorough treatment and reveal some additional properties. Namely, for a weak topology determined by a right ideal I of M , we show that the category of sheaves associated to this topology is a subtopos of M S e t (the presheaves over M) and determine the Lawvere–Tierney topology yielding the same subtopos, which is the Lawvere–Tierney topology associated to the idempotent hull of the (not necessarily idempotent) closure operator associated to I. We will then find conditions under which the subcategory of separated objects turns out to be a topos, and in the last section, we find conditions under which the category of sheaves becomes a De Morgan topos. [ABSTRACT FROM AUTHOR]

Let R be a ring and let n be an arbitrary but fixed positive integer. We characterize those rings R whose elements a satisfy at least one of the relations that a n + a or a n − a is a nilpotent whenever n ∈ ℕ ∖ { 1 }. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with x n − x nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with x n − x nilpotent, J. Algebra Appl. 21 (2022)], respectively. [ABSTRACT FROM AUTHOR]

Let (X ,) be a topological space. By the Skula topology (or the b -topology) on X , we mean the topology b () on X with basis the collection of all -locally closed sets of X , the resulting space (X , b ()) will be denoted by b (X). We show that the following results hold: (1) b (X) is an Alexandroff space if and only if the T 0 -reflection T 0 (X) of X is a T D -space. (2) b (X) is a Noetherian space if and only if T 0 (X) is finite. (3) If we denote by X * the Alexandroff extension of X , then b (X *) = (b (X)) * if and only if (X ,) is a Noetherian quasisober space. We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, b (b (b (X))) = b (b (X)). A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology Clop () of : The indiscrete components of Clop (X) are of the form C x = { x } ¯ ∩ (x) , where x ∈ X and (x) is the intersection of all open sets of X containing x (equivalently, C x = { y ∈ X : { x } ¯ = { y } ¯ }). We show that Clop (X) = b (b (X)) [ABSTRACT FROM AUTHOR]

Automorphic loops or A -loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring (Q , + , [. ,. ]) we can define an operation (∗) such that (Q , ∗) is an A -loop. We call it Lie automorphic loop. A half-isomorphism f : G → K between multiplicative systems G and K is a bijection from G onto K such that f (a b) ∈ { f (a) f (b) , f (b) f (a) } for any a , b ∈ G. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc.8 (1957) 1141–1144] that if G is a group then f is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism. [ABSTRACT FROM AUTHOR]

We prove a conjecture proposed in [A. De Sole, V. G. Kac and D. Valeri, Finite W -algebras for 𝔤 𝔩 N , Adv. Math. 327 (2018) 173–224.] describing the Lax type operator L (z) for the quantum finite W -algebras of 𝔤 𝔩 N in terms of a PBW generating system for the W -algebra. In doing so, we extend this result to an arbitrary good grading and an arbitrary isotropic subspace of 𝔤 [ 1 2 ]. [ABSTRACT FROM AUTHOR]

To unify different definitions of smash Hom-products in a Hom-bialgebra H , we firstly introduce the notion of 𝜃 -twisted smash Hom-product B # 𝜃 H. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product B # 𝜃 H and the twisted smash Hom-coproduct B × ω H to afford a Hom-bialgebra, which generalize the well-known Radford's biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math.137 (2014) 43–65]. Furthermore, we introduce the notion of the category of ϖ -Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math.137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the (𝜃 , ω) -twisted Radford's Hom-biproduct B × ω # 𝜃 H is a Hom-bialgebra if and only if (B , α B) is a Hom-bialgebra in the category of 𝜃 − 1 ω − 1 α H 2 -Yetter–Drinfeld modules H H ℋ 𝒴 𝒟 𝜃 − 1 ω − 1 α H − 2 , generalizing the well-known Majid's conclusion. [ABSTRACT FROM AUTHOR]

Let (𝒜 , - ⊗ -) be a symmetric monoidal closed exact category. This category is a natural framework to define the notions of purity and flatness. When 𝒜 is endowed with an injective cogenerator with respect to the exact structure, we show that an object ℱ in 𝒜 is flat if and only if any conflation ending in ℱ is pure. Furthermore, we prove a generalization of the Lambek Theorem (J. Lambek, A module is flat if and only if its character module is injective, Canad. Math. Bull.7 (1964) 237–243) in 𝒜. In the case 𝒜 is a quasi-abelian category, we prove that 𝒜 has enough pure injective objects. [ABSTRACT FROM AUTHOR]

Using the equivalence of unique generation and cleanness of C (X) , we give affirmative answers to questions raised in [I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc.  66 (1949) 464–491] and [D.D. Anderson et al., When are associates unit multiples? Rocky Mountain J. Math.  34 (2004) 811–828] for rings of real-valued continuous functions. In fact, we show that if C (X) is UG (uniquely generated) then M n (C (X)) is too, and R is strongly associate if and only if R [ [ x ] ] is, where R = C (X). We give topological characterizations of UG and AS (annihilator-stable) elements of C (X) for continuum spaces X and using this, we observe that the product of two UG elements need not be UG. It is shown that the set of elements in C (X) which have stable range 1 and the set of AS elements of C (X) coincide and several examples are given which show that the set of UG elements, the set of AS elements and the set of clean elements of C (X) can differ. Finally, we characterize spaces X for which every clean element of C (X) or every element of C (X) which has stable range 1 is UG. [ABSTRACT FROM AUTHOR]

For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is a maximal Cohen-Macaulay approximation of the trivial module and is the tensor identity of the stable category of Gorenstein-projective modules over the category algebra. We give conditions on when the trivial module is Gorenstein-projective. [ABSTRACT FROM AUTHOR]

The least number of products of linear forms that may be added together to obtain a given form is the Chow rank of this form. The Chow rank of a generic form corresponds to the smallest for which the th secant variety of the Chow variety fills the ambient space. We show that, except for certain known exceptions, this secant variety has the expected dimension for low values of . [ABSTRACT FROM AUTHOR]