2012/08/27 by Alborz Azarang, A. Azarang, Azarang, A.
Mathematics · #13B02 #13E05 #13G05 #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #math.RA #msc:13B02 #msc:13E05 #msc:13G05
paper · pdf · doi:10.48550/arxiv.1208.5298
arxiv created 2012/08/27 · openalex publication_date 2012/08/27 · arxiv updated 2012/08/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is proved that if D is a UFD and R is a D-algebra, such that U(R)∩ D≠ U(D), then R has a maximal subring. In particular, if R is a ring which either contains a unit x which is not algebraic over the prime subring of R, or R has zero characteristic and there exists a natural number n>1 such that (1)/(n)∈ R, then R has a maximal subring. It is shown that if R is a reduced ring with |R|>2^2ℵ0 or J(R)≠ 0, then any R-algebra has a maximal subring. Residually finite rings without maximal subrings are fully characterized. It is observed that every uncountable UFD has a maximal subring. The existence of maximal subrings in a noetherian integral domain R, in relation to either the cardinality of the set of divisors of some of its elements or the height of its maximal ideals, is also investigated.