Abstract

A ring extension is a ring homomorphism preserving identities. In this paper, we give the definition of relatively free modules on ring extensions and develop some basic properties of relatively free modules. Then we establish the relationship between relatively free modules and relatively projective modules. In particular, we prove that the relatively free modules, relatively projective modules and relatively injective modules on the ring extension $S\hookrightarrow S[x]/(x^n)$ coincide with $n\ge 2$ being a natural number, and that every such module has the form $S[x]/(x^n){\otimes }_{S}N$ or ${\mathrm{}\text{Hom}}_S(S[x]/(x^n),N)$ with N an S-module.

中文翻译：

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环形扩展模块上相对免费的模块

摘要

环扩展是环同态保留身份。在本文中，我们给出了环扩展上相对自由模块的定义，并开发了相对自由模块的一些基本性质。然后我们建立相对自由模块和相对投影模块之间的关系。特别地，我们证明了环扩展上的相对自由模、相对射影模和相对射模$秒\hookrightarrow 秒[X]/(X^n)$ 不谋而合 $n\ge 2$ 是一个自然数，并且每个这样的模块都有形式 $秒[X]/(X^n){\otimes }_{秒}N$ 或者 ${\mathrm{}\text{坎}}_{秒}(秒[X]/(X^n),N)$与N一个S模块。