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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 $n>1$.
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참고문헌 (14)
참고문헌 신청
D. D. Anderson / 2008 / Generalizations of prime ideals / Comm. Algebra 36(2):686~696
D. D. Anderson / 2003 / Weakly prime ideals / Houston J. Math 29(4):831~840
D. F. Anderson / 2011 / On n-absorbing ideals of commutative rings / Comm. Algebra 39(5):1646~1672
A. Badawi / 2007 / On 2-absorbing ideals of commutative rings / Bull. Austral. Math. Soc 75(3):417~429
A. Badawi / 2013 / On weakly 2-absorbing ideals of commutative rings / Houston J. Math 39(2):441~452
D. D. Anderson / 2003 / Weakly prime ideals / Houston J. Math 29(4):831~840
D. F. Anderson / 2011 / On n-absorbing ideals of commutative rings / Comm. Algebra 39(5):1646~1672
A. Badawi / 2007 / On 2-absorbing ideals of commutative rings / Bull. Austral. Math. Soc 75(3):417~429
A. Badawi / 2013 / On weakly 2-absorbing ideals of commutative rings / Houston J. Math 39(2):441~452
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