Translator Disclaimer
January 2020 The Complexity of Radicals and Socles of Modules
Huishan Wu
Notre Dame J. Formal Logic 61(1): 141-153 (January 2020). DOI: 10.1215/00294527-2019-0036

Abstract

This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of Z-modules, where the radical of a Z-module M is defined as the intersection of pM={px:xM} with p taken from all primes. It shows that ACA0 is equivalent to the existence of radicals of Z-modules over RCA0. We then study socles of modules over commutative rings with identity. The socle of an R-module M is the largest semisimple submodule of M. We show that the existence of socles of modules over a commutative ring with identity is equivalent to ACA0 over RCA0. Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.

Citation

Download Citation

Huishan Wu. "The Complexity of Radicals and Socles of Modules." Notre Dame J. Formal Logic 61 (1) 141 - 153, January 2020. https://doi.org/10.1215/00294527-2019-0036

Information

Received: 14 April 2018; Accepted: 15 March 2019; Published: January 2020
First available in Project Euclid: 29 November 2019

zbMATH: 07196096
MathSciNet: MR4054249
Digital Object Identifier: 10.1215/00294527-2019-0036

Subjects:
Primary: 03B30
Secondary: 03D15

Rights: Copyright © 2020 University of Notre Dame

JOURNAL ARTICLE
13 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.61 • No. 1 • January 2020
Back to Top