The Computational Complexity of Subclasses of Semiperfect Rings
This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semipe...
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| Vydané v: | Mathematics (Basel) Ročník 12; číslo 22; s. 3608 |
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| Jazyk: | English |
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MDPI AG
01.11.2024
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| ISSN: | 2227-7390, 2227-7390 |
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| Abstract | This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ20-hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π20-hard within the index set of computable rings. Finally, based on the Π20 definition of local rings, computable semiperfect rings can be described by Σ30 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ20-hard and Π20-hard within the index set of computable rings. |
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| AbstractList | This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ[sub.2] [sup.0]-hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π[sub.2] [sup.0]-hard within the index set of computable rings. Finally, based on the Π[sub.2] [sup.0] definition of local rings, computable semiperfect rings can be described by Σ[sub.3] [sup.0] formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ[sub.2] [sup.0]-hard and Π[sub.2] [sup.0]-hard within the index set of computable rings. This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ20-hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π20-hard within the index set of computable rings. Finally, based on the Π20 definition of local rings, computable semiperfect rings can be described by Σ30 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ20-hard and Π20-hard within the index set of computable rings. This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ 20 -hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π 20 -hard within the index set of computable rings. Finally, based on the Π 20 definition of local rings, computable semiperfect rings can be described by Σ 30 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ 20 -hard and Π 20 -hard within the index set of computable rings. |
| Audience | Academic |
| Author | Wu, Huishan |
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| Cites_doi | 10.1007/978-3-642-59971-2 10.1215/00294527-2019-0036 10.1007/978-3-642-31933-4 10.1007/978-3-031-11367-3 10.1215/00294527-2023-0003 10.1016/j.jalgebra.2008.06.007 10.1016/j.jalgebra.2009.07.039 10.1017/S0004972700031014 10.1007/978-1-4419-8616-0 10.1016/j.jalgebra.2012.09.020 10.1017/bsl.2014.32 10.2178/bsl/1146620059 10.1090/proc/12509 10.1093/acprof:oso/9780199230761.001.0001 10.1090/S0002-9947-2014-06115-1 10.1016/j.jalgebra.2007.02.058 10.1007/978-3-662-02460-7 10.1215/00294527-2021-0017 10.1016/j.apal.2022.103089 10.1007/978-1-4612-4418-9 |
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| SubjectTerms | Algebra Complexity computability theory computational complexity local ring Mathematics Rings (mathematics) semiperfect ring semisimple ring Sums |
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| Title | The Computational Complexity of Subclasses of Semiperfect Rings |
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