2024/09/12 by Alborz Azarang, Azarang, Alborz
Mathematics · #16K20 #16K40 #16S85 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2410.09051
openalex publication_date 2024/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The structure and the existence of maximal subrings in division rings are investigated. We see that if R is a maximal subring of a division ring D with center F and N(R)≠ U(R)∪ \0\, where N(R) is the normalizer of R in D, then either R is a division ring with [D:R]l=[D:R]r is finite or R is an Ore G-domain with certain properties. In particular, if F\subsetneq CD(R), the centralizer of R in D, then R=CD(β) is a division ring, for each β∈ CR(R)∖ F, [D:R]l is finite if and only if β is algebraic over F, [D:R]l=[D:R]r=[F[β]:F] and CR(R)=F[β]. On the other hand if R does not contains F, then R∩ F=CR(R) is a maximal subring of F. Consequently, if a division ring D has a noncentral element which is algebraic over the center of D, then D has a maximal subring. In particular, we prove that if D is a non-commutative division ring with center F, then either D has a maximal subring or dimF(D)≥ |F|. We study when a maximal subring of a division ring is a left duo ring or certain valuation rings. Finally, we prove that if D is an existentially complete division ring over a field K, then D has a maximal subring of the form CD(x) where D is finite over it. Moreover, if R is a maximal subring of D with K\subsetneq CR(R), then R=CD(x) for some x∈ D∖ K, which is algebraic over K.