An ensemble of idempotent lifting hypotheses
| Autor: | Pace P. Nielsen, Tsit Yuen Lam, Dinesh Khurana |
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| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Class (set theory) Algebra and Number Theory Ideal (set theory) Mathematics::Commutative Algebra Modulo 010102 general mathematics Structure (category theory) 010103 numerical & computational mathematics Characterization (mathematics) 01 natural sciences Lift (mathematics) Algebra symbols.namesake Idempotence symbols 0101 mathematics Mathematics Von Neumann architecture |
| Zdroj: | Journal of Pure and Applied Algebra. 222:1489-1511 |
| ISSN: | 0022-4049 |
| DOI: | 10.1016/j.jpaa.2017.07.008 |
| Popis: | Lifting idempotents modulo ideals is an important tool in studying the structure of rings. This paper lays out the consequences of lifting other properties modulo ideals, including lifting of von Neumann regular elements, lifting isomorphic idempotents, and lifting conjugate idempotents. Applications are given for IC rings, perspective rings, and Dedekind-finite rings, which improve multiple results in the literature. We give a new characterization of the class of exchange rings; they are rings where regular elements lift modulo all left ideals. We also uncover some hidden connections between these lifting properties. For instance, if regular elements lift modulo an ideal, then so do isomorphic idempotents. The converse is true when units lift. The logical relationships between these and several other important lifting properties are completely characterized. Along the way, multiple examples are developed that illustrate limitations to the theory. |
| Databáze: | OpenAIRE |
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