This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of -modules, where the radical of a -module is defined as the intersection of with taken from all primes. It shows that is equivalent to the existence of radicals of -modules over . We then study socles of modules over commutative rings with identity. The socle of an -module is the largest semisimple submodule of . We show that the existence of socles of modules over a commutative ring with identity is equivalent to over . 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.
"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