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A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8] and Nicholson [11] from projective semiperfect modules to arbitrary semiperfect modules. [ABSTRACT FROM AUTHOR]

Let M be a hypermodule over a hyperring R such that the intersection of any two subhypermodules of M, is a subhypermodule of M. We introduce the concept of an essential subhypermodule in M relative to an arbitrary subhypermodule T of M, which is called a T-essential subhypermodule of M. Our main goal in this work is to investigate properties of (relative) essential subhypermodules. We apply this concept to introduce extending hypermodules. Examples are provided to illustrate different concepts. [ABSTRACT FROM AUTHOR]

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra𝔢(2). [ABSTRACT FROM PUBLISHER]

The rate of a graded moduleMover a standard graded algebraR,rateR(M), is a measure of the growth of the shifts in the minimal graded free resolution ofM. In this paper, we study the rate of Veronese modules ofM. More precisely, it is shown thatfor allc≥1. This extends a result of Aramova et al. [2] As a consequence of this, ifMis generated in degree zero, then, for allc≥max{rateR(M),Rate(R)}. Also, it is shown that, for all integerss,c≥1. In particular case, we give a simple proof to a result of Backelin. [ABSTRACT FROM PUBLISHER]

In this paper, we study a new Lie superalgebra constructed by a 2|2-dimensional Balinsky–Novikov superalgebra, which is called the superalgebra ofW(2,2). It can be realized from semi product of theW-algebraW(2,2) and its module of the intermediate series. Finally, we determine all modules of the intermediate series over this superalgebra. Since it is difficult to do so directly, we make it by using modules of the intermediate series over the trivial super extension of the Witt algebra. [ABSTRACT FROM PUBLISHER]

In this paper, using Larsson’s functor with irreducible𝔰𝔩2-modulesV, we construct a class ofℤ2-graded modules for the Virasoro-like algebra and itsq-analogs. We determine the irreducibility of these modules for finite-dimensional or infinite-dimensionalVusing a unified method. In particular, these modules provide new irreducible weight modules with infinite-dimensional weight spaces for the corresponding algebras. [ABSTRACT FROM PUBLISHER]

Motivated by a well-known theorem of Mathieu’s on Harish–Chandra modules over the Virasoro algebra and its super version, we show that an irreducible quasifinite module over two classes of Lie superalgebras𝒮(q) of Block type is either a highest or lowest weight module or else a module of the intermediate series ifq≠−1. For such a module over𝒮(−1), we give a rough classification. [ABSTRACT FROM PUBLISHER]

Let R be a commutative Noetherian ring. In this article, we provide some new criteria for a semidualizing module to be dualizing in terms of special homological properties of module categories. The purpose of this article is twofold: first, it aims at improving Christensen's and Takahashi et al.'s characterizations of dualizing modules; secondly, while applying these criteria to the ring itself, we not only recover some results of Jenda and Xu, respectively, but also obtain a new characterization of Gorenstein rings. [ABSTRACT FROM PUBLISHER]

We give several characterizations of Zimmermann's uniform chain condition on matrix subgroups. In particular, we prove that a family of modules satisfies it precisely when the direct sum of their pure injective envelopes splits in their direct product. We also prove that a weaker non-uniform version of this condition is equivalent to Σ-algebraic compactness of reduced products. [ABSTRACT FROM PUBLISHER]

Let X ⊂ (H0(L)) be a smooth projective variety embedded by the complete linear system associated to a very ample line bundle L on X. We call [image omitted] the section module of L. It has been known that the syzygies of RL as R = Sym(H0(L))-module play important roles in understanding geometric properties of X [2, 3, 5, 9, 10] even if X is not projectively normal. Generalizing the case of N2, p [2, 10], we prove some uniform theorems on higher normality and syzygies of a given linearly normal variety X and general inner projections when RL satisfies property N3, p (Theorems 1.1, 1.2, and Proposition 3.1). In particular, our uniform bounds are sharp as hyperelliptic curves and elementary transforms of elliptic ruled surfaces show. [ABSTRACT FROM AUTHOR]

This work examines the possible projectivity of 2-modular block parts of nonprojective Lefschetz characters over 2-local geometries of several sporadic groups. Previously known results on M12, J2, and HS are mentioned for completeness. The main new results are on the sporadic groups Suz, Co3, Ru, O'N, and He. For each group, the Lefschetz character is calculated, and its 2-modular block parts are examined for projectivity. In each case it is confirmed that a nonprincipal block part contains a nonprojective summand. The case of O'N is additionally found to have a nonprojective summand in its principal block part. Nineteen of the sporadic groups (including many previously known cases) are categorized into three classes based on projectivity properties of their Lefschetz characters. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]

Recently Jain, López-Permouth and Syed classified indecomposable non-local right continuous right π-rings. In the present article we describe the structure of right continuous right π-rings. [ABSTRACT FROM AUTHOR]