[Paper Review] Cancellation properties in ideal systems: an $\boldsymbol{e.a.b.}$ not $\boldsymbol{a.b.}$ star operation

Marco Fontana, K. Alan Loper+1 more|arXiv (Cornell University)|Feb 11, 2010

Rings, Modules, and Algebras3 references3 citations

TL;DR

This paper demonstrates that Gilmer's e.a.b. (endlich arithmetisch brauchbar) cancellation condition is strictly weaker than Krull's a.b. (arithmetisch brauchbar) condition for star operations on integral domains. By modifying a construction from a prior example, the authors exhibit a star operation that satisfies e.a.b. cancellation but fails a.b. cancellation, proving the two conditions are not equivalent in general.

ABSTRACT

We show that Krull's exttt{a.b.} cancellation condition is a properly stronger condition than Gilmer's exttt{e.a.b.} cancellation condition for star operations.

Motivation & Objective

• To clarify the logical relationship between two cancellation conditions in multiplicative ideal theory: Krull's a.b. and Gilmer's e.a.b.
• To resolve ambiguity in prior literature suggesting e.a.b. might imply a.b. for star operations.
• To construct a counterexample showing that e.a.b. does not imply a.b. for star operations.
• To demonstrate that the $t$-operation and related star operations do not fully capture the e.a.b. condition in certain non-Noetherian domains.
• To clarify the distinction between $t$-operation and a newly constructed star operation that is e.a.b. but not a.b.

Proposed method

• Construct a non-Noetherian UFD $D = k[X_1,X_2,...]_N$ localized at the maximal ideal $N = (X_1,X_2,...)$.
• Define a set $\mathcal{J}$ of fractional ideals generated by $xF^t$, $yM$, and $zM^2$ for $x,y,z \in K \setminus \{0\}$ and $F \in \boldsymbol{f}(D)$.
• Use $\mathcal{J}$ to define a star operation $\ast$ via $E^\ast = \bigcap \{ J \in \mathcal{J} \mid J \supseteq E \}$.
• Show that $F^\ast = F^t$ fails for certain $F \in \boldsymbol{f}(D)$ with $F^t = D$, such as $I = (X_i,X_j)$, by proving $I^\ast = M$.
• Use the $b$-Kronecker function ring $\mathrm{Kr}(D,b)$ to analyze the behavior of ideals under $\ast$, leveraging its Bézout domain structure.
• Derive a contradiction assuming $I \not\subseteq G^\ast$ for $G \in \boldsymbol{f}(D)$, showing that $I \subseteq G^\ast$ must hold, thus proving $\ast$ is e.a.b.

Experimental results

Research questions

• RQ1 Is Gilmer's e.a.b. cancellation condition strictly weaker than Krull's a.b. cancellation condition for star operations on integral domains?
• RQ2 Can a star operation be e.a.b. without being a.b.?
• RQ3 Does the $t$-operation fully capture the e.a.b. property in non-Noetherian domains?
• RQ4 Can a star operation be constructed such that $F^\ast = F^t$ fails for some $F \in \boldsymbol{f}(D)$ with $F^t = D$?
• RQ5 Is there a domain where $I^\ast = M$ for $I = (X_i,X_j)$ with $I^t = D$, showing $\ast$ is not $t$-like?

Key findings

• The star operation $\ast$ constructed in the paper satisfies the e.a.b. condition but fails the a.b. condition.
• For $I = (X_i,X_j)$ with $i \neq j$, $I^\ast = M$, while $I^t = D$, proving $F^\ast \neq F^t$ for some $F \in \boldsymbol{f}(D)$.
• The operation $\ast$ is e.a.b. because any inclusion $(FG)^\ast \subseteq (FH)^\ast$ implies $G^\ast \subseteq H^\ast$ for $F,G,H \in \boldsymbol{f}(D)$.
• The operation $\ast$ is not a.b. because there exist $F,G,H \in \boldsymbol{F}(D)$ such that $(FG)^\ast \subseteq (FH)^\ast$ but $G^\ast \not\subseteq H^\ast$, as shown via contradiction in the Kronecker function ring.
• The contradiction arises from assuming $M\mathrm{Kr}(D,b) \subseteq \mathcal{H}$ for a principal ideal $\mathcal{H}$, leading to $X_n \in \varphi\mathrm{Kr}(D,b)$ for all $n$, which is impossible if $\varphi$ involves only finitely many indeterminates.
• The construction confirms that the e.a.b. condition is strictly weaker than the a.b. condition, resolving a foundational question in multiplicative ideal theory.