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Let $D$ be an integral domain and $S$ a multiplicative subset of $D.$ An ascending chain $(I_k)_{k\in\mathbb{N}}$ of ideals of $D$ is said to be $S$-stationary if there exist a positive integer $n$ and an $s\in S$ such that for each $k\geq n,$ $sI_k\subseteq I_n.$ As a generalization of domains satisfying ACCP (resp., ACC on $*$-ideals) we define $D$ to satisfy $S$-ACCP (resp., $S$-ACC on $*$-ideals) if every ascending chain of principal ideals (resp., $*$-ideals) of $D$ is $S$-stationary. One of main results of this paper is the Hilbert basis theorem for an integral domain satisfying $S$-ACCP. Also we investigate the class of such domains $D$ and we generalize some known related results in the literature. Finally some illustrative examples regarding the introduced concepts are given.
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참고문헌 (17)
참고문헌 신청
Hamed Ahmed / 2015 / S-Noetherian rings of the forms A[X] and A / Communications in Algebra 43(9):3848~3856
Hamed Ahmed / 2015 / Modules Satisfying the S-Noetherian Property and S-ACCR / Communications in Algebra 44(5):1941~1951
D.D. Anderson / 1988 / Some remarks on star operations and the class group / Journal of Pure and Applied Algebra 51(1-2):27~33
D.D. Anderson / 1990 / Factorization in integral domains / Journal of Pure and Applied Algebra 69(1):1~19
D.D. Anderson / 1991 / Splitting the t-class group / Journal of Pure and Applied Algebra 74(1):17~37
Hamed Ahmed / 2015 / Modules Satisfying the S-Noetherian Property and S-ACCR / Communications in Algebra 44(5):1941~1951
D.D. Anderson / 1988 / Some remarks on star operations and the class group / Journal of Pure and Applied Algebra 51(1-2):27~33
D.D. Anderson / 1990 / Factorization in integral domains / Journal of Pure and Applied Algebra 69(1):1~19
D.D. Anderson / 1991 / Splitting the t-class group / Journal of Pure and Applied Algebra 74(1):17~37
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