Amount algebras

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 Prüfer domain or a torsion-free valuation ring and $I$ is a radical ideal of $R$, then ${\omega }_{R[][X]]}(I[[X]])={\omega }_R(I)$.

Comments welcome! 9 pages!