Algebra Colloquium

For exceptional Lie superalgebras D(2,1;α), G(3) and F(4), Gröbner-Shirshov bases and the corresponding super-Lyndon-Shirshov monomial bases are constructed.

In this paper, we establish a structure theorem of naturally ordered rpp semigroups with max-idempotents. Results on naturally ordered semigroups with max-idempotents in the literature are extended and amplified. Some previous results obtained by Blyth, McFadden, Guo and Xie on regular and abundant ordered semigroups are extended to ordered rpp semigroups.

In this paper we characterize all the lexsegment ideals which are normally torsion-free. This will provide a large class of normally torsion-free monomial ideals which are not square-free. Our characterization is given in terms of the ends of the lexsegment. We also prove that, for lexsegment ideals, the property being normally torsion-free is equivalent to the property of the depth function being constant.

In this paper, we obtain (non-commutative) Gröbner-Shirshov bases for rectangular bands, small extensions of semigroups and Bruck-Reilly extensions of monoids in terms of the general product.

Let M be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Spec^s(M) of second submodules of M and the spectrum Spec^c(M) of coprime submodules of M, study several properties of these spaces and investigate their interplay with the algebraic properties of M.

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete characterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

We study the structure of graded Leibniz algebras with arbitrary dimension and over an arbitrary base field 𝕂. We show that any of such algebras 𝔏 with a symmetric G-support is of the form with U a subspace of 𝔏₁, the homogeneous component associated to the unit element 1 in G, and any I_j a well described graded ideal of 𝔏, satisfying [I_j, I_k]=0 if j ≠ k. In the case of 𝔏 being of maximal length, we characterize the gr-simplicity of the algebra in terms of connections in the support of the grading.

Let X be a zero-dimensional scheme in ℙ¹ × ℙ¹. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depend only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.

The Albert algebra and the Jordan algebra of a nondegenerate symmetric bilinear form are shown to be zero product determined.

We investigate the structure of rings over which every finitely generated δ-supplemented module is supplemented. Some characterizations of this type of rings are given. We establishe some properties of ⊕-δ-supplemented modules. It is showed that for a ring R, finitely generated δ-supplemented R-modules are supplemented if and only if finitely generated ⊕-δ-supplemented R-modules are ⊕-supplemented.

Let R=⊕_{n ≥ 0} R_n be a standard graded ring, 𝔞 ⊇ ⊕_{n > 0} R_n an ideal of R, and M, N two finitely generated graded R-modules. This paper studies the homogeneous components of graded generalized local cohomology modules. We show that for any i ≥ 0, the n-th graded component of the i-th generalized local cohomology module of M and N with respect to 𝔞 vanishes for all n ≫ 0. Some sufficient conditions are proposed to satisfy the equality . Also, some sufficient conditions are proposed for the tameness of such that or i=cd_𝔞(M,N), where and cd_𝔞(M,N) denote the R₊-finiteness dimension and the cohomological dimension of M and N with respect to 𝔞, respectively. Finally, we consider the Artinian property of some submodules and quotient modules of , where j is the first or last non-minimax level of .

In the present paper we describe a class of Φ-Auslander-Yoneda algebras over K[x]/(xⁿ) in terms of quivers with relations, and prove that they are actually cellular algebras in the sense of Graham and Lehrer.

Let R be a prime ring and n > 1 be a fixed positive integer. If g is a nonzero generalized derivation of R such that g(x)ⁿ=g(x) for all x ∈ R, then R is commutative except when R is a subring of the 2 × 2 matrix ring over a field. Moreover, we generalize the result to the case g(f(x_i))ⁿ = g(f(x_i)) for all x₁, x₂, …, x_t∈ R, where f(X_i) is a multilinear polynomial.

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection. It is obvious that ∗-clean rings are clean. Vaš asked whether there exists a clean ring with involution that is not ∗-clean. In this paper, we investigate when a group ring RG is ∗-clean, where ∗ is the classical involution on RG. We obtain necessary and sufficient conditions for RG to be ∗-clean, where R is a commutative local ring and G is one of the groups C₃, C₄, S₃ and Q₈. As a consequence, we provide many examples of group rings which are clean but not ∗-clean.