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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| description | 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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| spelling | doaj-art-9d0a4b6241db43c2a6f88ead30d1a1b32024-11-26T18:12:01ZengMDPI AGMathematics2227-73902024-11-011222360810.3390/math12223608The Computational Complexity of Subclasses of Semiperfect RingsHuishan Wu0School of Information Science, Beijing Language and Culture University, 15 Xueyuan Road, Haidian District, Beijing 100083, ChinaThis 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.https://www.mdpi.com/2227-7390/12/22/3608computability theorycomputational complexitysemisimple ringlocal ringsemiperfect ring |
| spellingShingle | Huishan Wu The Computational Complexity of Subclasses of Semiperfect Rings Mathematics computability theory computational complexity semisimple ring local ring semiperfect ring |
| title | The Computational Complexity of Subclasses of Semiperfect Rings |
| title_full | The Computational Complexity of Subclasses of Semiperfect Rings |
| title_fullStr | The Computational Complexity of Subclasses of Semiperfect Rings |
| title_full_unstemmed | The Computational Complexity of Subclasses of Semiperfect Rings |
| title_short | The Computational Complexity of Subclasses of Semiperfect Rings |
| title_sort | computational complexity of subclasses of semiperfect rings |
| topic | computability theory computational complexity semisimple ring local ring semiperfect ring |
| url | https://www.mdpi.com/2227-7390/12/22/3608 |
| work_keys_str_mv | AT huishanwu thecomputationalcomplexityofsubclassesofsemiperfectrings AT huishanwu computationalcomplexityofsubclassesofsemiperfectrings |