In this paper, we deal with the level simplicial complexes of codimension two. We give a structural characterization for the Stanley-Reisner rings of these complexes and a numerical classification for their h-vectors. We prove that for a sequence h=(h0,h1,h2,& mldr;,hs) of integers with h0=1 and h1=2 , the following conditions are equivalent: (1) h is a pure O-sequence, (2) h is a shellable O-sequence, (3) h is the h-vector of a level complex, (4) h is the h-vector of a matroid complex. These equivalent conditions yield a new proof of a conjecture due to Stanley concerning pure O-sequences and a positive answer to a refinement of the conjecture due to Chari, both in the case of codimension two.
Assume that R is an associative ring with identity. An R -module X is called c -injective if, for every closed submodule K of every R -module M , every homomorphism from K to X extends to M . In this paper, we show that a strong Mori domain D is a Krull domain if and only if every w -simple D -module is c -injective. As a result, over a Krull domain, if M is a direct product of w -simple modules, then any homomorphism from a closed submodule K of M to M can be lifted to M .
This paper is devoted to the study of dimonoids, i.e., algebraic structures equipped with two associative binary operations satisfying a given system of axioms. We introduce several new classes of dimonoids and determine their automorphism groups and halos. In particular, we construct examples of iso-dual nonabelian dimonoids, as well as nonabelian dimonoids with nonempty halo. Using these constructions, we obtain a complete classification, up to isomorphism, of all nonabelian noncommutative nontrivial dimonoids of order 3, thereby resolving the problem concerning this classification. Consequently, all three-element dimonoids are classified up to isomorphism, yielding exactly 52 pairwise nonisomorphic dimonoids of order 3. Finally, we present the results of computer computations determining the numbers of all pairwise nonisomorphic dimonoids of orders up to 5, as well as all pairwise nonisomorphic commutative, abelian, and rectangular dimonoids of orders up to 6, obtained using GAP, Python, and C++.
Let M denote the vector space of 2 & times; 2 matrices with coefficients in F-3 and trace zero. Let G = SL2(F-3). Then G acts on M via conjugation. Let R = (S(M*) (R) A(M*)) be the algebra of differential forms on M. We compute a minimal generating set for R-G as a commutative-graded algebra. In doing so we utilize the theory of Cohen-Macaulay modules and results in the theory of covariants.
In this paper, we introduce the notion of algebraic & lowast;-Ricci solitons of three-dimensional contact Lie groups. We give the classification of algebraic & lowast;-Ricci solitons of three-dimensional unimodular Lie groups and prove that the algebraic & lowast;-Ricci solitons of three-dimensional non-unimodular Lie groups are steady or expanding. We also give an example of expanding algebraic & lowast;-Ricci soliton to illustrate the application of the theorem.
New criteria for a left hereditary ring to be left perfect and right coherent are found. One of them requires any left module over the ring to have a unique up to isomorphism decomposition into a direct sum of a projective submodule and a stable submodule. The essential ingredient in the proof of this criterion is a purely categorical result-a certain form of the well-known epireflective subcategory theorem. Applying the dual of this theorem, we give also a new proof of Zheng-Xu He's structure theorem for left modules over left hereditary left Noetherian rings.
Let G be a finite group. A subgroup H of G is said to be s-quasinormal in G if H permutes with every Sylow subgroup of G. A subgroup H of G is said to be s-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-quasinormal subgroup of G. A subgroup H of G is said to be E-supplemented in G if there is a subgroup K of G such that G=HK and H boolean AND K <= HeG, where HeG is the subgroup of H generated by all those subgroups of H which are s-quasinormally embedded in G. Let P be a Sylow p-subgroup of a finite group G and let p be the smallest prime dividing the order of G. In this paper, we investigate the p-nilpotence of G under the assumption that the second maximal subgroups of P are E-supplemented in G. We extend and improve some known results.
