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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.00539 |
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| _version_ | 1866915132848734208 |
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| author | Safaeipour, Mahboubeh Moghimi, Hosein Fazaeli Rashedi, Fatemeh |
| author_facet | Safaeipour, Mahboubeh Moghimi, Hosein Fazaeli Rashedi, Fatemeh |
| contents | Let $R$ be a commutative ring with identity, and let $\R(R)$ denote the semiring of radical ideals of $R$. The radical functor $\R$, from the category of $R$-modules $R{-}\boldsymbol{\sf{Mod}}$ to the category of $\R(R)$-semimodules $\R(R){-}\boldsymbol{\sf{Semod}}$, maps any complex $\M=(M_n, f_n)_{n\geq 0}$ of $R$-modules to a complex $\R(\M)=(\R(M_n), \R(f_n))_{n\geq 0}$ of $\R(R)$-semimodules, where $\R(M_n)$ consists of radical submodules of $M_n$, and the $\R(R)$-semimodule homomorphisms $\R(f_n):\R(M_n)\rightarrow \R(M_{n-1})$ are defined by $\R(f_n)(N)=\rad(f_n(N))$. The $n$-th radical homology of the complex $(\R(M_n), \R(f_n))_{n\geq 0}$, denoted $H_n(\R(\M))$, consists of radical submodules $N$ of $M_n$ such that $f_n(N)$ is contained in the radical of the zero submodule of $M_{n-1}$, and two such radical submodules are equivalent under the Bourne relation modulo the image of $\R(f_{n+1})$. $H_n(\R(-))$ is regarded as a covariant functor from the category $\boldsymbol{\sf{Ch}}(R{-}\boldsymbol{\sf{Mod}})$ of chain complexes of $R$-modules to $\R(R){-}\boldsymbol{\sf{Semod}}$, which acts identically on any pair of homotopic maps of complexes of $R$-modules. In particular, if $\M$ and $\M'$ are homotopically equivalent, then $H_n(\R(\M))$ and $H_n(\R(\M'))$ are isomorphic $\R(R)$-semimodules. We provide conditions under which $H_n(\R(-))$ induces a long exact sequence of radical homology modules for any short exact sequence of complexes of $R$-modules, and satisfies the naturality condition for exact homology sequences. Finally, we introduce a projective resolution for an $R$-module $M$ based on $\R(R)$-semimodules and give conditions under which such a projective resolution exists and is unique up to a homotopy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00539 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Homology Theory for the Semimodules of Radical Submodules Safaeipour, Mahboubeh Moghimi, Hosein Fazaeli Rashedi, Fatemeh Commutative Algebra Let $R$ be a commutative ring with identity, and let $\R(R)$ denote the semiring of radical ideals of $R$. The radical functor $\R$, from the category of $R$-modules $R{-}\boldsymbol{\sf{Mod}}$ to the category of $\R(R)$-semimodules $\R(R){-}\boldsymbol{\sf{Semod}}$, maps any complex $\M=(M_n, f_n)_{n\geq 0}$ of $R$-modules to a complex $\R(\M)=(\R(M_n), \R(f_n))_{n\geq 0}$ of $\R(R)$-semimodules, where $\R(M_n)$ consists of radical submodules of $M_n$, and the $\R(R)$-semimodule homomorphisms $\R(f_n):\R(M_n)\rightarrow \R(M_{n-1})$ are defined by $\R(f_n)(N)=\rad(f_n(N))$. The $n$-th radical homology of the complex $(\R(M_n), \R(f_n))_{n\geq 0}$, denoted $H_n(\R(\M))$, consists of radical submodules $N$ of $M_n$ such that $f_n(N)$ is contained in the radical of the zero submodule of $M_{n-1}$, and two such radical submodules are equivalent under the Bourne relation modulo the image of $\R(f_{n+1})$. $H_n(\R(-))$ is regarded as a covariant functor from the category $\boldsymbol{\sf{Ch}}(R{-}\boldsymbol{\sf{Mod}})$ of chain complexes of $R$-modules to $\R(R){-}\boldsymbol{\sf{Semod}}$, which acts identically on any pair of homotopic maps of complexes of $R$-modules. In particular, if $\M$ and $\M'$ are homotopically equivalent, then $H_n(\R(\M))$ and $H_n(\R(\M'))$ are isomorphic $\R(R)$-semimodules. We provide conditions under which $H_n(\R(-))$ induces a long exact sequence of radical homology modules for any short exact sequence of complexes of $R$-modules, and satisfies the naturality condition for exact homology sequences. Finally, we introduce a projective resolution for an $R$-module $M$ based on $\R(R)$-semimodules and give conditions under which such a projective resolution exists and is unique up to a homotopy. |
| title | A Homology Theory for the Semimodules of Radical Submodules |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2502.00539 |