Journal of the
Korean Mathematical Society JKMS

Let $R$ be a commutative ring with $1\neq 0$ and $n$ a positive integer. In this article, we introduce the $n$-Krull dimension of $R$, denoted $\dim_n R$, which is the supremum of the lengths of chains of $n$-absorbing ideals of $R$. We study the $n$-Krull dimension in several classes of commutative rings. For example, the $n$-Krull dimension of an Artinian ring is finite for every positive integer $n$. In particular, if $R$ is an Artinian ring with $k$ maximal ideals and $l(R)$ is the length of a composition series for $R$, then $\dim_n R = l(R) -k $ for some positive integer $n$. It is proved that a Noetherian domain $R$ is a Dedekind domain if and only if $\dim_nR=n$ for every positive integer $n$ if and only if $\dim_2R=2$. It is shown that Krull's (Generalized) Principal Ideal Theorem does not hold in general when prime ideals are replaced by $n$-absorbing ideals for some $n>1$.

Keywords: $n$-absorbing ideal, $n$-Krull dimension, $n$-height, Artinian ring, Dedekind domain

MSC numbers: Primary 13A15; Secondary 13C15, 13E10, 13F05