• Title, Summary, Keyword: exact sequences

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ESSENTIAL EXACT SEQUENCES

  • Akray, Ismael;Zebari, Amin
    • Communications of the Korean Mathematical Society
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    • v.35 no.2
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    • pp.469-480
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    • 2020
  • Let R be a commutative ring with identity and M a unital R-module. We give a new generalization of exact sequences called e-exact sequences. A sequence $0{\rightarrow}A{\longrightarrow[20]^f}B{\longrightarrow[20]^g}C{\rightarrow}0$ is said to be e-exact if f is monic, Imf ≤e Kerg and Img ≤e C. We modify many famous theorems including exact sequences to one includes e-exact sequences like 3 × 3 lemma, four and five lemmas. Next, we prove that for torsion-free module M, the contravariant functor Hom(-, M) is left e-exact and the covariant functor M ⊗ - is right e-exact. Finally, we define e-projective module and characterize it. We show that the direct sum of R-modules is e-projective module if and only if each summand is e-projective.

ON QUASI-EXACT SEQUENCES

  • ANVARIYEH, S.M.;DAVVAZ, B.
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.1
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    • pp.149-155
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    • 2005
  • The notion of U-exact sequence (or quasi-exact sequence) of modules was introduced by Davvaz and Parnian-Garamaleky as a generalization of exact sequences. In this paper, we prove further results about quasi-exact sequences. In particular, we give a generalization of Schanuel's Lemma. Also we obtain some relation-ship between quasi-exact sequences and superfluous (or essential) submodules.

RELATIONS OF SHORT EXACT SEQUENCES CONCERNING AMALGAMATED FREE PRODUCTS

  • Shin, Woo Taeg
    • Korean Journal of Mathematics
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    • v.14 no.2
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    • pp.217-226
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    • 2006
  • In this paper, we investigate the mutual relation among short exact sequences of amalgamated free products which involve augmentation ideals and relation modules. In particular, we find out commutative diagrams having a steady structure in the sense that all of their three columns and rows are short exact sequences.

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ON DOUBLE EXACT SEQUENCES

  • Myung, Sung
    • Korean Journal of Mathematics
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    • v.16 no.2
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    • pp.123-134
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    • 2008
  • In the present article, we give a description of a chain complex of a cyclic cohomology type proposed by Grayson, which is constructed from the double exact sequences of an exact category. We also provide some results which relate it with motivic cohomology of a smooth scheme over a field.

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Clustered Segment Index for Efficient Approximate Searching on the Secondary Structure of Protein Sequences (클러스터 세그먼트 인덱스를 이용한 단백질 이차 구조의 효율적인 유사 검색)

  • Seo Min-Koo;Park Sang-Hyun;Won Jung-Im
    • Journal of KIISE:Databases
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    • v.33 no.3
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    • pp.251-260
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    • 2006
  • Homology searching on the primary structure (i.e., amino acid arrangement) of protein sequences is an essential part in predicting the functions and evolutionary histories of proteins. However, proteins distant in an evolutionary history do not conserve amino acid residue arrangements, while preserving their structures. Therefore, homology searching on proteins' secondary structure is quite important in finding out distant homology. In this manuscript, we propose an indexing scheme for efficient approximate searching on the secondary structure of protein sequences which can be easily implemented in RDBMS. Exploiting the concept of clustering and lookahead, the proposed indexing scheme processes three types of secondary structure queries (i.e., exact match, range match, and wildcard match) very quickly. To evaluate the performance of the proposed method, we conducted extensive experiments using a set of actual protein sequences. CSI was proved to be faster than the existing indexing methods up to 6.3 times in exact match, 3.3 times in range match, and 1.5 times in wildcard match, respectively.