Amount algebras

TL;DR

This paper introduces amount algebras as a generalization of content algebras and proves a formula relating the smallest n-absorbing number of ideals in these algebras to that in the base ring under certain conditions.

Contribution

It generalizes the concept of content algebras to amount algebras and establishes a formula connecting their n-absorbing properties with those of the base ring.

Findings

Proves the formula $ ext{omega}_B(I^{ ext{epsilon}})= ext{omega}_R(I)$ under certain conditions.

Shows the formula holds for Pr"ufer domains and torsion-free valuation rings with radical ideals.

Extends the understanding of n-absorbing ideals in algebraic structures.

Abstract

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson-Badawi $\omega_{R[X]}(I[X])=\omega_R(I)$ conjecture, we prove that under some conditions, the formula $\omega_B(I^{\epsilon})=\omega_R(I)$ holds for some amount $R$-algebras $B$ and some ideals $I$ of $R$, where $\omega_R(I)$ is the smallest positive integer $n$ that the ideal $I$ of $R$ is $n$-absorbing. A corollary to the mentioned formula is that if, for example, $R$ is a Pr\"{u}fer domain or a torsion-free valuation ring and $I$ is a radical ideal of $R$, then $\omega_{R[][X]]}(I[[X]])=\omega_R(I)$.