Relatively free modules on ring extensions

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.

Keywords:

• Frobenius extensions
• relatively free modules
• relatively projective modules
• ring extensions

2020 Mathematics Subject Classification:

• 18E30
• 16D10
• 18G25

Acknowledgements

The authors thank the referee for helpful comments and suggestions.

Additional information

Funding

The research work is partially supported by the Guangxi Natural Science Foundation Project (No. 2018GXNSFBA281150) and the Guangxi Natural Science Foundation Project (No. 2018GXNSFAA138191).