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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| Published in: | Mathematics (Basel) Vol. 12; no. 22; p. 3608 |
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| Format: | Journal Article |
| Language: | 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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| References | Downey (ref_8) 2008; 320 Conidis (ref_14) 2021; 62 ref_21 Melnikov (ref_11) 2014; 20 ref_20 Downey (ref_7) 2014; 366 ref_1 ref_3 ref_2 Conidis (ref_13) 2009; 322 Downey (ref_12) 2007; 314 ref_16 ref_15 Downey (ref_6) 2013; 373 Wu (ref_18) 2022; 173 Lempp (ref_9) 1997; 56 Wu (ref_19) 2023; 64 Riggs (ref_10) 2015; 143 ref_4 Calvert (ref_5) 2006; 12 Wu (ref_17) 2020; 61 |
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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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