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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/22/3608 |
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| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="sans-serif">Σ</mi><mn>2</mn><mn>0</mn></msubsup></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="sans-serif">Π</mi><mn>2</mn><mn>0</mn></msubsup></semantics></math></inline-formula>-hard within the index set of computable rings. Finally, based on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="sans-serif">Π</mi><mn>2</mn><mn>0</mn></msubsup></semantics></math></inline-formula> definition of local rings, computable semiperfect rings can be described by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="sans-serif">Σ</mi><mn>3</mn><mn>0</mn></msubsup></semantics></math></inline-formula> formulas. As a corollary, we find that the index set of computable semiperfect rings can be both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="sans-serif">Σ</mi><mn>2</mn><mn>0</mn></msubsup></semantics></math></inline-formula>-hard and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="sans-serif">Π</mi><mn>2</mn><mn>0</mn></msubsup></semantics></math></inline-formula>-hard within the index set of computable rings. |
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| ISSN: | 2227-7390 |