Certain towers of ramified minimal ring extensions of commutative rings

ABSTRACT

Let p be a prime number. Let B be the idealization ℤ(+)ℤ∕pℤ and N: = pℤ(+)ℤ∕pℤ. Let B⊂R be a ramified (integral minimal) ring extension, with crucial maximal ideal 𝒩 and (necessarily) an element y such that R = B[y], y²∈B, y³∈B and y𝒩⊆𝒩. Then B is the only (commutative unital) ring properly contained between ℤ and R if and only if 𝒩 = N and either $\{y^2,y^3\}⊈\mathbb{Z}$ or py∉ℤ. Moreover, there are exactly three isomorphism classes of such rings R. Let A: = ℤ∕p^αℤ for some integer α≥2. Let A⊂D⊂E be a tower of ramified ring extensions, (necessarily) with an element z such that E = D[z], z²∈D, z³∈D and zℳ⊆ℳ, where ℳ is the crucial maximal ideal of D⊂E. Then D is the only ring properly contained between A and E if and only if either z²∉A or pz∉A. There are at least two, but only finitely many, isomorphism classes of such rings E. We complete the characterization of the rings with exactly two proper (unital) subrings, including a classification up to isomorphism of these rings in the case of characteristic 0.

KEYWORDS:

• Commutative ring
• crucial maximal ideal
• idealization
• integral domain
• integral
• minimal ring extension
• ramified ring extension

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 13B99
• Secondary 13B21, 13A15, 13G05

Acknowledgment

The author wishes to thank the referee for a meticulous critique and for finding an error in the originally submitted proof of Lemma 2.18(b).