Let $R\ $be a commutative ring with nonzero identity and $I\ $a proper ideal of $R.\ $Then $I\ $is called a uniformly $pr$-ideal if there exists $N\in% \mathbb{N} $ such that $ab\in I\ $with $ann(a)=0\ $then $b^{N}\in I.\ $We say that the smallest $N\in% \mathbb{N} $ is called order of $I\ $and denoted by $ord_{R}(I)=N.\ $In this paper, we give some examples and characterizations of this new class of ideals.
📌 3 Key Takeaways
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Objective & Hypothesis:Let $R\ $be a commutative ring with nonzero identity and $I\ $a proper ideal of $R.\ $Then $I\ $is called a uniformly $pr$-ideal if there exists $N\in% \mathbb{N} $ such that $ab\in I\ $with $ann(a)=0\ $then $b^{N}\in I.\ $We say that the smallest $N\in% \mathbb{N} $ is called order of $I\ $and denoted by $ord_{R}(I)=N.\ $In this paper, we give some examples and characterizations of this new class of ideals.
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Key Findings & Data:Let $R\ $be a commutative ring with nonzero identity and $I\ $a proper ideal of $R.\ $Then $I\ $is called a uniformly $pr$-ideal if there exists $N\in% \mathbb{N} $ such that $ab\in I\ $with $ann(a)=0\ $then $b^{N}\in I.\ $We say that the smallest $N\in% \mathbb{N} $ is called order of $I\ $and denoted by $ord_{R}(I)=N.\ $In this paper, we give some examples and characterizations of this new class of ideals.
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Conclusion & Impact:Let $R\ $be a commutative ring with nonzero identity and $I\ $a proper ideal of $R.\ $Then $I\ $is called a uniformly $pr$-ideal if there exists $N\in% \mathbb{N} $ such that $ab\in I\ $with $ann(a)=0\ $then $b^{N}\in I.\ $We say that the smallest $N\in% \mathbb{N} $ is called order of $I\ $and denoted by $ord_{R}(I)=N.\ $In this paper, we give some examples and characterizations of this new class of ideals.
Let $R\ $be a commutative ring with nonzero identity and $I\ $a proper ideal of $R.\ $Then $I\ $is called a uniformly $pr$-ideal if there exists $N\in% \mathbb{N} $ such that $ab\in I\ $with $ann(a)=0\ $then $b^{N}\in I.\ $We say that the smallest $N\in% \mathbb{N} $ is called order of $I\ $and denoted by $ord_{R}(I)=N.\ $In this paper, we give some examples and characterizations of this new class of ideals.
ÜREGEN, R. N. (2019). On uniformly $pr$-ideals in commutative rings. Turkish Journal of Mathematics, Cilt: 43 - Sayı: 4, 1878–1886. https://ajindex.com/articles/208975