A classification of the minimal ring extensions of certain commutative rings

Abstract

All rings considered are commutative with identity and all ring extensions are unital. Let R be a ring with total quotient ring T . The integral minimal ring extensions of R are catalogued via generator-and-relations. If T is von Neumann regular and no maximal ideal of R is a minimal prime ideal of R, the minimal ring extensions of R are classified, up to R-algebra isomorphism, as the minimal overrings (within T ) of R and, for maximal ideals M of R, the idealizations R(+)R/M and the direct products R × R/M. If T is von Neumann regular, the minimal ring extensions of R in which R is integrally closed are characterized as certain overrings, up to R-algebra isomorphism, in terms of Kaplansky transforms and divided prime ideals, generalizing work of Ayache on integrally closed domains; no restriction on T is needed if R is quasilocal. One application generalizes a recently announced result of Picavet and Picavet-L'Hermitte on the minimal overrings of a local Noetherian ring. Examples are given to indicate sharpness of the results.

Modules With Finitely Many Submodules

“…Note that other papers by D. E. Dobbs [6], and D. E. Dobbs with P.-J. Cahen, T. G. Lucas [5], J. Shapiro [9], B. Mullins and ourselves [7] also went against the same trend. It is worth noticing here that FCP extensions of integral domains are (ignoring fields) extensions of overrings as a quick look at [5,Theorems 4.1,4.4] shows because FCP extensions are composites of finitely many minimal extensions.…”

“…Note that other papers by D. E. Dobbs [6], and D. E. Dobbs with P.-J. Cahen, T. G. Lucas [5], J. Shapiro [9], B. Mullins and ourselves [7] M is finitely generated, R/C has finitely many ideals and M P is cyclic for any prime ideal P of R containing C such that R/P is infinite.…”

The Space of Maximal Subrings of a Commutative Ring

“…We remind the reader that if S is a maximal subring of a ring R, then the ring extension S ⊆ R is called a minimal ring extension. This kind of ring extensions are studied first in [15,17,18], and recently in [11][12][13], see also [34]. We summarize some historical and well-known facts from the latter references in Section 1.…”

Maximal Subrings and Covering Numbers of Finite Semisimple Rings

“…There are many papers in the literature studying maximal subalgebras of an algebra A over a field F (see e.g. [9,10,11,21,22,23] and the references therein), and for commutative rings the related notion of minimal ring extensions has become popular (for this, the papers [6,7,8] are a good starting point). However, less work has been done on classifying the maximal subrings of a general algebra A .…”