在σ[M]中的降秩
Reduced rank in $σ[M]$
AI总结：
本文研究了在σ[M]中降秩的概念扩展，探讨了商范畴的谱性质，并给出了模块商在特定torsion理论下的有限长度条件，以及推广Small定理的条件。
摘要：
我们利用Raggi等人引入的素子模的概念，将降秩的概念扩展到σ[M]的模论背景下。当M是一个半素Goldie模块时，我们研究σ[M]模由M的M-内射包络所生成的hereditary torsion理论的商范畴，并证明该商范畴是谱的。接着，我们考虑在σ[M]中由M/素 radical(M)的M-内射包络所生成的hereditary torsion理论，并确定当模块的商（相对于此torsion理论）在商范畴中具有有限长度时的条件。最后，我们给出关于具有内射环S的模块M的条件，使得S是Artinian环中的一个order，从而推广Small定理。
英文摘要：
Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context of $σ[M]$. We study the quotient category of $σ[M]$ modulo the hereditary torsion theory cogenerated by the $M$-injective hull of $M$, when $M$ is a semiprime Goldie module. We prove that this quotient category is spectral. We then consider the hereditary torsion theory in $σ[M]$ cogenerated by the $M$-injective hull of $M/\mathfrak{L}(M)$, where $\mathfrak{L}(M)$ is the prime radical of $M$, and we determine when the module of quotients of $M$, with respect to this torsion theory, has finite length in the quotient category. Finally, we give conditions on a module $M$ with endomorphism ring $S$ under which $S$ is an order in an Artinian ring, extending Small's Theorem.
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