On $n$-absorbing ideals and the $n$-Krull dimension of a commutative ring
1
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
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 $n>1$.
Abstract
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 $n>1$.
참고문헌 (14)
- D. D. Anderson, "Generalizations of prime ideals", Comm. Algebra, 36(2), pp.686~696 (2008)
- D. D. Anderson, "Weakly prime ideals", Houston J. Math, 29(4), pp.831~840 (2003)
- D. F. Anderson, "On n-absorbing ideals of commutative rings", Comm. Algebra, 39(5), pp.1646~1672 (2011)
- A. Badawi, "On 2-absorbing ideals of commutative rings", Bull. Austral. Math. Soc, 75(3), pp.417~429 (2007)
- A. Badawi, "On weakly 2-absorbing ideals of commutative rings", Houston J. Math, 39(2), pp.441~452 (2013)
- Ayman Badawi, "On 2-absorbing primary ideals in commutative rings", 대한수학회보, 51(4), pp.1163~1173 (2014), 대한수학회
- Ayman Badawi, "ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS", 대한수학회지, 52(1), pp.97~111 (2015), 대한수학회
- A. Yousefian Darani, "On 2-absorbing preradicals", J. Algebra Appl, 14, pp.22~ (2015)
- A. Yousefian Darani, "On 2-absorbing commutative semigroups and their applications to rings", Semigroup Forum, 86(1), pp.83~91 (2013)
- M. Ebrahimpour, "On generalizations of prime ideals", Comm. Algebra, 40(4), pp.1268~1279 (2012)
- D. Eisenbud, "Commutative Algebra, Graduate Texts in Mathematics", pp.150~ (1995), SpringerVerlag
- M. D. Larson, "Multiplicative Theory of Ideals" (1971), Academic Press
- H. Matsumura, "Commutative Ring Theory" (1989), Cambridge University Press
- R. Y. Sharp, "Steps in Commutative Algebra" (1990), Cambridge University Press