Algebraic Structures and Clean Ring Theory
Summary
Algebraic structures in ring theory form the backbone of modern algebra, encompassing rings, modules and related constructs that encode operations of addition and multiplication. Within this broad domain, clean ring theory emerges as a specialised study of rings in which every element admits a decomposition as a sum of a unit and an idempotent. This property enables a finer understanding of ring composition, revealing internal symmetries and facilitating module classification. Clean ring theory generalises concepts from regular and π-regular rings and investigates extensions such as nil-cleanness, bi-cleanness in Hopf modules and monochromatic conditions in finite rings. Developments in this field have linked cleanness to matrix decompositions over diverse domains, combinatorial colouring problems and coalgebra modules, establishing interconnections with representation theory and computational algebra. Practical applications span cryptographic protocols, coding theory and algorithmic realisation of algebraic processes, where explicit idempotent–unit decompositions underpin efficient computation and structural analysis.
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Algebraic Structures and Clean Ring Theory publication trend
The graph below shows the total number of articles in algebraic structures and clean ring theory across all publications each year (not limited to Nature Index journals).
Technical terms
Clean ring: A ring in which every element can be expressed as the sum of a unit and an idempotent.
Idempotent: An element e of a ring satisfying e² = e.
Nilpotent: An element n of a ring such that nᵏ = 0 for some positive integer k.
Unit: An invertible element in a ring that possesses a multiplicative inverse.
Hopf module: A module that is simultaneously a comodule over a bialgebra, with compatible action and coaction structures.
References
- Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence. Linear Algebra and its Applications (2023).
- 2-Products of Idempotent by Nilpotent Matrices. Bulletin of the Iranian Mathematical Society (2024).
- Bi-clean and clean Hopf modules. AIMS Mathematics (2022).
- On Monochromatic Clean Condition on Certain Finite Rings. Mathematics (2023).
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