A Homology Theory for the Semimodules of Radical Submodules
Mahboubeh Safaeipour, Hosein Fazaeli Moghimi, Fatemeh Rashedi
TL;DR
The paper develops a homology theory for semimodules of radical submodules by applying the radical functor $\\mathcal{R}$ to chain complexes over a commutative ring $R$, producing complexes of $\\mathcal{R}(R)$-semimodules and defining radical homology $H_n(\\mathcal{R}(\\mathcal{M}))$ via $Z_n(\\mathcal{R}(\\mathcal{M}))$ and $B_n(\\mathcal{R}(\\mathcal{M}))$. It proves functoriality, homotopy invariance, and natural long exact sequences for short exact sequences of complexes under suitable conditions, and introduces radical projective resolutions with existence and lifting properties up to homotopy. The results extend homological algebra concepts to semimodules over radical semirings, yielding a framework for exact sequences, acyclicity criteria, and resolution theory in this context. The work has potential implications for studying radical submodule structures and their homological behavior in algebraic settings lacking classical module theory completeness.
Abstract
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.