Computably Enumerable Semisimple Rings

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

Mathematics, 2025, vol. 13, issue 3, 1-23

Abstract: The theory of semisimple rings plays a fundamental role in noncommutative algebra. We study the complexity of the problem of semisimple rings using the tools of computability theory. Following the general idea of computably enumerable (c.e. for short) universal algebras, we define a c.e. ring as the quotient ring of a computable ring modulo a c.e. congruence relation and view such rings as structures in the language of rings, together with a binary relation. We formalize the problem of being semisimple for a c.e. ring by the corresponding index set and prove that the index set of c.e. semisimple rings is Σ 3 0 -complete. This reveals that the complexity of the definability of c.e. semisimple rings lies exactly in the Σ 3 0 of the arithmetic hierarchy. As applications of the complexity results on semisimple rings, we also obtain the optimal complexity results on other closely connected classes of rings, such as the small class of finite direct products of fields and the more general class of semiperfect rings.

Keywords: computability theory; universal algebra; arithmetic hierarchy; computably enumerable ring; semisimple ring (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/13/3/337/pdf (application/pdf)
https://www.mdpi.com/2227-7390/13/3/337/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:13:y:2025:i:3:p:337-:d:1572735

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().