On the Commutative Rings with At Most Two Proper Subrings

Abstract

The commutative rings with exactly two proper (unital) subrings are characterized. An initial step involves the description of the commutative rings having only one proper subring.

Characterizing finite Boolean rings by using finite chains of subrings

“…Then Z/q α j j Z is a special principal ideal ring (SPIR), but not a field. Hence, the assertions about the decomposed or inert candidates for B j follow from [5,Propositions 10 and 8]. Finally (if α j ≥ 2), the "finite and at least 2" assertion about the ramified candidates for B j follows from [7, Theorem 3.4 (f)], while the "as small as 2" and "larger than 2" assertions follow from parts (e) and (f), respectively, of [5, Proposition 12].…”

On minimal ring extensions of finite rings

“…It follows from the above comments that every commutative minimal ring extension of a finite (commutative) ring must be finite (cf. also [7,Proposition 7]). The first classification result on minimal ring extensions was due to Ferrand-Olivier [18, Lemme 1.2]: if k is a field, then a nonzero commutative k-algebra B is a minimal ring extension of k (when we view k ⊆ B via the injective structural map k → B) if and only if B is k-algebra isomorphic to (exactly one of) k[X]/(X 2 ), k × k or a minimal field extension of k. Now let R ⊂ S be an integral ring extension with S commutative, and consider its conductor M := (R : S) (:= {s ∈ S | sS ⊆ R}).…”