Abstract
These are the notes of a non-standard course of Algebra. It deals with elementary theory of commutative monoids and non-commutative rings. Most of what is taught in a master course of Commutative Algebra holds not only for commutative rings, but more generally for any commutative monoid, which shows that the additive group structure on a commutative ring has little importance.
In the rest of the notes of the course presented here, we introduce the basic notions of non-commutative rings and their modules, stressing the difference with what happens in the case of commutative rings.
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Acknowledgements
The author is partially supported by Ministero dell’Istruzione, dell’Università e della Ricerca (Progetto di ricerca di rilevante interesse nazionale “Categories, Algebras: Ring-Theoretical and Homological Approaches (CARTHA)”), Fondazione Cariverona (Research project “Reducing complexity in algebra, logic, combinatorics - REDCOM” within the framework of the programme “Ricerca Scientifica di Eccellenza 2018”), and Dipartimento di Matematica “Tullio Levi-Civita” of Università di Padova (Research program DOR1828909 “Anelli e categorie di moduli”).
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Facchini, A. (2021). Commutative Monoids, Noncommutative Rings and Modules. In: Clementino, M.M., Facchini, A., Gran, M. (eds) New Perspectives in Algebra, Topology and Categories. Coimbra Mathematical Texts, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-84319-9_3
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