2024/06/04 by Alborz Azarang, Azarang, Alborz
Mathematics · #16D25 #16D80 #16N60 #16P20 #16P40 #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2406.12890
openalex publication_date 2024/06/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a maximal subring of a ring T, and (R:T), (R:T)l and (R:T)r denote the greatest ideal, left ideal and right ideal of T which are contained in R, respectively. It is shown that (R:T)l and (R:T)r are prime ideals of R and |MinR((R:T))|≤ 2. We prove that if TR has a maximal submodule, then (R:T)l is a right primitive ideal of R. We investigate that when (R:T)r is a completely prime (right) ideal of R or T. If R is integrally closed in T, then (R:T)l and (R:T)r are prime one-sided ideals of T. We observe that if (R:T)lT=T, then T is a finitely generated left R-module and (R:T)l is a finitely generated right R-module. We prove that Char(R/(R:T)l)=Char(R/(R:T)r), and if Char(T) is neither zero or a prime number, then (R:T)≠ 0. If |Min(R)|≥ 3, then (R:T) and (R:T)l(R:T)r are nonzero ideals. Finally we study the Noetherian and the Artinian properties between R and T.