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Let $R$ be a commutative ring with identity, and $\phi : \mathscr{I}(R) \rightarrow \mathscr{I}(R) \cup \{\varnothing\}$ be a function where $\mathscr{I}(R)$ is the set of all ideals of $R$. Following \cite{ab}, a proper ideal $P$ of $R$ is called a $\phi$-prime ideal if $x, y \in R$ with $xy\in P-\phi(P)$ implies $x\in P$ or $y\in P$. For an ideal $I$ of $R$, we define the $\phi$-radical $\sqrt[\phi]{I}$ to be the intersection of all $\phi$-prime ideals of $R$ containing $I$, and show that this notion inherits most of the essential properties of the usual notion of radical of an ideal. We also investigate when the set of all $\phi$-prime ideals of $R$, denoted $\operatorname{Spec}_{\phi}(R)$, has a Zariski topology analogous to that of the prime spectrum $\operatorname{Spec}(R)$, and show that this topological space is Noetherian if and only if $\phi$-radical ideals of $R$ satisfy the ascending chain condition.
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참고문헌 (9)
참고문헌 신청
A. G. Agarg¨un / 1999 / Unique factorization rings with zero divisors / Comm. Algebra 27(4):1967~1974
D. D. Anderson / 2008 / Generalization of prime ideals / Comm. Algebra 36(2):686~696
D. D. Anderson / 2003 / Weakly prime ideals / Houston J. Math 29(4):831~840
D. D. Anderson / 2009 / Idealization of a module / J. Commut. Algebra 1(1):3~56
S. M. Bhatwadekar / 2005 / Unique factorization and birth of almost primes / Comm. Algebra 33(1):43~49
D. D. Anderson / 2008 / Generalization of prime ideals / Comm. Algebra 36(2):686~696
D. D. Anderson / 2003 / Weakly prime ideals / Houston J. Math 29(4):831~840
D. D. Anderson / 2009 / Idealization of a module / J. Commut. Algebra 1(1):3~56
S. M. Bhatwadekar / 2005 / Unique factorization and birth of almost primes / Comm. Algebra 33(1):43~49
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