• Title, Summary, Keyword: Noetherian ring

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ON FINITENESS PROPERTIES ON ASSOCIATED PRIMES OF LOCAL COHOMOLOGY MODULES AND EXT-MODULES

  • Chu, Lizhong;Wang, Xian
    • Journal of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.239-250
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    • 2014
  • Let R be a commutative Noetherian (not necessarily local) ring, I an ideal of R and M a finitely generated R-module. In this paper, by computing the local cohomology modules and Ext-modules via the injective resolution of M, we proved that, if for an integer t > 0, dim$_RH_I^i(M){\leq}k$ for ${\forall}i$ < t, then $$\displaystyle\bigcup_{i=0}^{j}(Ass_RH_I^i(M))_{{\geq}k}=\displaystyle\bigcup_{i=0}^{j}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$$ for ${\forall}j{\leq}t$ and ${\forall}n$ >0. This shows that${\bigcup}_{n>0}(Ass_RExt_R^i(R/I^n,M))_{{\geq}k}$ is a finite set for ${\forall}i{\leq}t$. Also, we prove that $\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1^{n_1},x_2^{n_2},{\ldots},x_i^{n_i})M)_{{\geq}k}=\displaystyle\bigcup_{i=1}^{r}(Ass_RM/(x_1,x_2,{\ldots},x_i)M)_{{\geq}k}$ if $x_1,x_2,{\ldots},x_r$ is M-sequences in dimension > k and $n_1,n_2,{\ldots},n_r$ are some positive integers. Here, for a subset T of Spec(R), set $T_{{\geq}i}=\{{p{\in}T{\mid}dimR/p{\geq}i}\}$.

ASSOCIATED PRIME SUBMODULES OF A MULTIPLICATION MODULE

  • Lee, Sang Cheol;Song, Yeong Moo;Varmazyar, Rezvan
    • Honam Mathematical Journal
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    • v.39 no.2
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    • pp.275-296
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    • 2017
  • All rings considered here are commutative rings with identity and all modules considered here are unital left modules. A submodule N of an R-module M is said to be extended to M if $N=aM$ for some ideal a of R and it is said to be fully invariant if ${\varphi}(L){\subseteq}L$ for every ${\varphi}{\in}End(M)$. An R-module M is called a [resp., fully invariant] multiplication module if every [resp., fully invariant] submodule is extended to M. The class of fully invariant multiplication modules is bigger than the class of multiplication modules. We deal with prime submodules and associated prime submodules of fully invariant multiplication modules. In particular, when M is a nonzero faithful multiplication module over a Noetherian ring, we characterize the zero-divisors of M in terms of the associated prime submodules, and we show that the set Aps(M) of associated prime submodules of M determines the set $Zdv_M(M)$ of zero-dvisors of M and the support Supp(M) of M.

COFINITENESS OF GENERAL LOCAL COHOMOLOGY MODULES FOR SMALL DIMENSIONS

  • Aghapournahr, Moharram;Bahmanpour, Kamal
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.5
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    • pp.1341-1352
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    • 2016
  • Let R be a commutative Noetherian ring, ${\Phi}$ a system of ideals of R and $I{\in}{\Phi}$. In this paper among other things we prove that if M is finitely generated and $t{\in}\mathbb{N}$ such that the R-module $H^i_{\Phi}(M)$ is $FD_{{\leq}1}$ (or weakly Laskerian) for all i < t, then $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all i < t and for any $FD_{{\leq}0}$ (or minimax) submodule N of $H^t_{\Phi}(M)$, the R-modules $Hom_R(R/I,H^t_{\Phi}(M)/N)$ and $Ext^1_R(R/I,H^t_{\Phi}(M)/N)$ are finitely generated. Also it is shown that if cd I = 1 or $dimM/IM{\leq}1$ (e.g., $dim\;R/I{\leq}1$) for all $I{\in}{\Phi}$, then the local cohomology module $H^i_{\Phi}(M)$ is ${\Phi}$-cofinite for all $i{\geq}0$. These generalize the main results of Aghapournahr and Bahmanpour [2], Bahmanpour and Naghipour [6, 7]. Also we study cominimaxness and weakly cofiniteness of local cohomology modules with respect to a system of ideals.

COLOCALIZATION OF LOCAL HOMOLOGY MODULES

  • Rezaei, Shahram
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.1
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    • pp.167-177
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    • 2020
  • Let I be an ideal of Noetherian local ring (R, m) and M an artinian R-module. In this paper, we study colocalization of local homology modules. In fact we give Colocal-global Principle for the artinianness and minimaxness of local homology modules, which is a dual case of Local-global Principle for the finiteness of local cohomology modules. We define the representation dimension rI (M) of M and the artinianness dimension aI (M) of M relative to I by rI (M) = inf{i ∈ ℕ0 : HIi (M) is not representable}, and aI (M) = inf{i ∈ ℕ0 : HIi (M) is not artinian} and we will prove that i) aI (M) = rI (M) = inf{rIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R)} ≥ inf{aIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R)}, ii) inf{i ∈ ℕ0 : HIi (M) is not minimax} = inf{rIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}. Also, we define the upper representation dimension RI (M) of M relative to I by RI (M) = sup{i ∈ ℕ0 : HIi (M) is not representable}, and we will show that i) sup{i ∈ ℕ0 : HIi (M) ≠ 0} = sup{i ∈ ℕ0 : HIi (M) is not artinian} = sup{RIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R)}, ii) sup{i ∈ ℕ0 : HIi (M) is not finitely generated} = sup{i ∈ ℕ0 : HIi (M) is not minimax} = sup{RIR𝖕 (𝖕M) : 𝖕 ∈ Spec(R) ∖ {𝔪}}.