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In this paper we construct a ring A which has annihilator condition (a.c.) and we show that neither nor has this property. This answers in negative a question asked by Hong, Kim, Lee and Nielsen. We also show that there is an algebra A which does not have annihilator condition while both and have this property. 相似文献
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We construct a minimal free resolution of the semigroup ring in terms of minimal resolutions of and when is a numerical semigroup obtained by gluing two numerical semigroups and . Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of , and prove that the minimal free resolution of has a differential graded algebra structure provided the resolutions of and possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in . 相似文献
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Alan Koch Timothy Kohl Paul J. Truman Robert Underwood 《Journal of Pure and Applied Algebra》2019,223(5):2230-2245
Let be a finite separable extension of fields whose Galois closure has Galois group G. Greither and Pareigis use Galois descent to show that a Hopf algebra giving a Hopf–Galois structure on has the form for some group N of order . We formulate criteria for two such Hopf algebras to be isomorphic as Hopf algebras, and provide a variety of examples. In the case that the Hopf algebras in question are commutative, we also determine criteria for them to be isomorphic as K-algebras. By applying our results, we complete a detailed analysis of the distinct Hopf algebras and K-algebras that appear in the classification of Hopf–Galois structures on a cyclic extension of degree , for p an odd prime number. 相似文献
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We provide a generalization of pseudo-Frobenius numbers of numerical semigroups to the context of the simplicial affine semigroups. In this way, we characterize the Cohen-Macaulay type of the simplicial affine semigroup ring . We define the type of S, , in terms of some Apéry sets of S and show that it coincides with the Cohen-Macaulay type of the semigroup ring, when is Cohen-Macaulay. If is a d-dimensional Cohen-Macaulay ring of embedding dimension at most , then . Otherwise, might be arbitrary large and it has no upper bound in terms of the embedding dimension. Finally, we present a generating set for the conductor of S as an ideal of its normalization. 相似文献
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Let be a polynomial ring, where is a field, and G be a simple graph on n vertices. Let be the vertex cover ideal of G. Herzog, Hibi and Ohsugi have conjectured that all powers of vertex cover ideals of chordal graph are componentwise linear. Here we establish the conjecture for the special case of trees. We also show that if G is a unicyclic vertex decomposable graph, then symbolic powers of are componentwise linear. 相似文献
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We explicitly determine generators of cyclic codes over a non-Galois finite chain ring of length , where p is a prime number and k is a positive integer. We completely classify that there are three types of principal ideals of and four types of non-principal ideals of , which are associated with cyclic codes over of length . We then obtain a mass formula for cyclic codes over of length . 相似文献
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《Discrete Mathematics》2022,345(10):112998
Let G be a graph and let f be a positive integer-valued function on . In this paper, we show that if for all , , then G has a spanning tree T containing an arbitrary given matching such that for each vertex v, , where denotes the number of components of and denotes the number of components of the induced subgraph with the vertex set S. This is an improvement of several results. Next, we prove that if for all , , then G admits a spanning closed walk passing through the edges of an arbitrary given matching meeting each vertex v at most times. This result solves a long-standing conjecture due to Jackson and Wormald (1990). 相似文献
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《Discrete Mathematics》2022,345(7):112866
Let G be a graph with n vertices. A path decomposition of G is a set of edge-disjoint paths containing all the edges of G. Let denote the minimum number of paths needed in a path decomposition of G. Gallai Conjecture asserts that if G is connected, then . If G is allowed to be disconnected, then the upper bound for was obtained by Donald [7], which was improved to independently by Dean and Kouider [6] and Yan [14]. For graphs consisting of vertex-disjoint triangles, is reached and so this bound is tight. If triangles are forbidden in G, then can be derived from the result of Harding and McGuinness [11], where g denotes the girth of G. In this paper, we also focus on triangle-free graphs and prove that , which improves the above result with . 相似文献
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《Discrete Mathematics》2022,345(5):112786
Let G be a connected graph with vertices and edges. The nullity of G, denoted by , is the multiplicity of eigenvalue zero of the adjacency matrix of G. Ma, Wong and Tian (2016) proved that unless G is a cycle of order a multiple of 4, where is the elementary cyclic number of G and is the number of leaves of G. Recently, Chang, Chang and Zheng (2020) characterized the leaf-free graphs with nullity , thus leaving the problem to characterize connected graphs G with nullity when . In this paper, we solve this problem completely. 相似文献
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Sanming Zhou 《Journal of Pure and Applied Algebra》2019,223(3):931-947
We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of , with connection sets and , respectively, where () is an mth primitive root of unity, A a nonzero ideal of , and ? Euler's totient function. We call them the mth cyclotomic graph and the second kind mth cyclotomic graph, and denote them by and , respectively. We give a necessary and sufficient condition for to be a perfect t-code in and a necessary condition for to be such a code in , where is an integer and D an ideal of containing A. In the case when , is known as an Eisenstein–Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for to be a perfect t-code in , where with β dividing α. In the literature such conditions are known to be sufficient when and under an additional condition. We give a classification of all first kind Frobenius circulants of valency 2p and prove that they are all pth cyclotomic graphs, where p is an odd prime. Such graphs belong to a large family of Cayley graphs that are efficient for routing and gossiping. 相似文献
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Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup of , in terms of a genetic basis of P. We also introduce a deflation map , for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of , when P is an elementary abelian p-group. 相似文献