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:x∈M} 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. (shrink)

Schur’s Lemma says that the endomorphism ring of a simple left R-module is a division ring. It plays a fundamental role to prove classical ring structure theorems like the Jacobson Density Theorem and the Wedderburn–Artin Theorem. We first define the endomorphism ring of simple left R-modules by their Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi ^{0}_{1}$$\end{document} subsets and show that Schur’s Lemma is provable in RCA0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm RCA_{0}$$\end{document}. A ring (...) R is left primitive if there is a faithful simple left R-module and left semisimple if the left regular module RR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_{R}R$$\end{document} is semisimple. The Jacobson Density Theorem and the Wedderburn-Artin Theorem characterize left primitive ring and left semisimple ring, respectively. We then study such theorems from the standpoint of reverse mathematics. (shrink)

This paper studies the structure of semisimple rings using techniques of reverse mathematics, where a ring is left semisimple if the left regular module is a finite direct sum of simple submodules. The structure theorem of left semisimple rings, also called Wedderburn-Artin Theorem, is a famous theorem in noncommutative algebra, says that a ring is left semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings. We provide a proof for the (...) theorem in $$\mathrm RCA_{0}$$, showing the structure theorem for computable semisimple rings. The decomposition of semisimple rings as finite direct products of matrix rings over division rings is unique. Based on an effective proof of the Jordan-Hölder Theorem for modules with composition series, we also provide an effective proof for the uniqueness of the matrix decomposition of semisimple rings in $$\mathrm RCA_{0}$$. (shrink)

Anderson and Csima defined a bounded jump operator for bounded-Turing reduction, and studied its basic properties. Anderson et al. constructed a low bounded-high set and conjectured that such sets cannot be computably enumerable. Ng and Yu proved that bounded-high c.e. sets are Turing complete, thus answered the conjecture positively. Wu and Wu showed that bounded-low sets can be superhigh by constructing a Turing complete bounded-low c.e. set. In this paper, we continue the study of the comparison between the bounded-jump and (...) Turing jump. We show that low c.e. sets are not all bounded-low and that incomplete superhigh c.e. sets can be bounded-low. (shrink)

This paper studies effective aspects of Jacobson radicals of rings and their applications from the viewpoint of reverse mathematics. First, we propose four radicals of rings, showing that the first order (resp., second order) left and right Jacobson radical coincide in (resp., ). Second, we study Jacobson radicals in left (resp., right) local rings and show that the second order left and right Jacobson radical of left (resp., right) local rings coincide within. Third, we apply our results about Jacobson radicals (...) of rings and local rings to study properties of left (resp., right) strongly indecomposable modules; furthermore, we study effective aspects of typical lemmas and theorems related to strongly indecomposable modules, and show that proves the Fitting Lemma as well as the Krull‐Schmidt‐Azumaya Theorem and that proves the Krull‐Schmidt Theorem. (shrink)

The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to over. We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to over. In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre‐orders to orders, where Zorn's lemma is used. We then prove that “pre‐orders (...) on rings not necessarily commutative extend to orders” is equivalent to. (shrink)

This paper studies various equivalent characterizations of left semisimple rings from the standpoint of reverse mathematics. We first show that \ is equivalent to the statement that any left module over a left semisimple ring is semisimple over \. We then study characterizations of left semisimple rings in terms of projective modules as well as injective modules, and obtain the following results: \ is equivalent to the statement that any left module over a left semisimple ring is projective over \; (...) \ is equivalent to the statement that any left module over a left semisimple ring is injective over \; \ proves the statement that if every cyclic left R-module is projective, then R is a left semisimple ring; \ proves the statement that if every cyclic left R-module is injective, then R is a left semisimple ring. (shrink)

This article studies the complexity of decomposability of rings from the perspective of computability. Based on the equivalence between the decomposition of rings and that of the identity of rings, we propose four kinds of rings, namely, weakly decomposable rings, decomposable rings, weakly block decomposable rings, and block decomposable rings. Let R be the index set of computable rings. We study the complexity of subclasses of computable rings, showing that the index set of computable weakly decomposable rings is m-complete Σ10 (...) within R and that the index set of computable decomposable (resp., weakly block decomposable, block decomposable) rings is m-complete Σ20 within R. (shrink)