The Computational Complexity of Subclasses of Semiperfect Rings

Huishan Wu ()
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Huishan Wu: School of Information Science, Beijing Language and Culture University, 15 Xueyuan Road, Haidian District, Beijing 100083, China

Mathematics, 2024, vol. 12, issue 22, 1-19

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 Σ 2 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 2 0 -hard within the index set of computable rings. Finally, based on the Π 2 0 definition of local rings, computable semiperfect rings can be described by Σ 3 0 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ 2 0 -hard and Π 2 0 -hard within the index set of computable rings.

Keywords: computability theory; computational complexity; semisimple ring; local ring; semiperfect ring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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