Let R be a commutative ring with identity and S a multiplicative subset of R. We introduce and study a generalization of composition series of modules, called u-S-composition series. Let M be an R-module. A finite chain of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(n + 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> submodules of M, <Equation ID="Equ1"> <EquationSource Format="TEX">$\begin{aligned} M_0=0\subseteq M_1\subseteq \cdots \subseteq M_n \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>M</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>⊆</mo> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>⊆</mo> <mo>⋯</mo> <mo>⊆</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>is called a u-S-composition series of length n for M provided that, for each <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$i\in \{1,2,\ldots , n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M_i/M_{i-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>i</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>M</mi> <mrow> <mi>i</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> </mrow> </math></EquationSource> </InlineEquation> is u-S-simple, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M/M_n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo stretchy="false">/</mo> <msub> <mi>M</mi> <mi>n</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> is u-S-torsion. We establish a u-S-counterpart of the Jordan–Hölder theorem, showing that any two u-S-composition series of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ M $</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> have the same length. Furthermore, we prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ M $</EquationSource> <EquationSource Format="MATHML"><math> <mi>M</mi> </math></EquationSource> </InlineEquation> has a u-S-composition series if and only if it is both u-S-Noetherian and u-S-Artinian. Some illustrative examples are provided to support our results.

On a generalization of composition series

期刊论文
发表日期：

2025年8月18日

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Ayoub Bouziri

摘要

Let R be a commutative ring with identity and S a multiplicative subset of R. We introduce and study a generalization of composition series of modules, called u-S-composition series. Let M be an R-module. A finite chain of $(n + 1)$ $(n + 1)$ submodules of M, $\begin{aligned} M_0=0\subseteq M_1\subseteq \cdots \subseteq M_n \end{aligned}$ $\begin{matrix} M_{0} = 0 \subseteq M_{1} \subseteq \dots \subseteq M_{n} \end{matrix}$ is called a u-S-composition series of length n for M provided that, for each $i\in \{1,2,\ldots , n\}$ $i \in {1, 2, \dots, n}$ , $M_i/M_{i-1}$ $M_{i} / M_{i - 1}$ is u-S-simple, and $M/M_n$ $M / M_{n}$ is u-S-torsion. We establish a u-S-counterpart of the Jordan–Hölder theorem, showing that any two u-S-composition series of $ M $ $M$ have the same length. Furthermore, we prove that $ M $ $M$ has a u-S-composition series if and only if it is both u-S-Noetherian and u-S-Artinian. Some illustrative examples are provided to support our results.