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Let $R$ be a commutative ring with $1\neq 0$. In this paper, we introduce a subclass of the class of $1$-absorbing primary ideals called the class of strongly $1$-absorbing primary ideals. A proper ideal $I$ of $R$ is called strongly $1$-absorbing primary if whenever nonunit elements $a, b, c \in R$ and $abc \in I$, then $ab \in I$ or $c \in \sqrt{0}$. Firstly, we investigate basic properties of strongly 1-absorbing primary ideals. Hence, we use strongly 1-absorbing primary ideals to characterize rings with exactly one prime ideal (the $UN$-rings) and local rings with exactly one non maximal prime ideal. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the prime ideals, the primary ideals and the 1-absorbing primary ideals. In the end of this paper, we give an idea about some strongly 1-absorbing primary ideals of the quotient rings, the polynomial rings, and the power series rings.
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참고문헌 (7)
참고문헌 신청
Ayman BADAWI / 2016 / Generalizations of 2-absorbing primaryideals of commutative rings / TURKISH JOURNAL OF MATHEMATICS 40:703~717
Ayman Badawi / 2014 / On 2-absorbing primary ideals in commutative rings / 대한수학회보 51(4):1163~1173
Ayman Badawi / 2015 / ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS / 대한수학회지 52(1):97~111
Ayman Badawi / 2020 / On 1-absorbing primary ideals of commutative rings / Journal of Algebra and Its Applications 19(06):2050111
J. W. Brewer / 1981 / Power Series over Commutative Rings / Marcel Dekker, Inc
Ayman Badawi / 2014 / On 2-absorbing primary ideals in commutative rings / 대한수학회보 51(4):1163~1173
Ayman Badawi / 2015 / ON WEAKLY 2-ABSORBING PRIMARY IDEALS OF COMMUTATIVE RINGS / 대한수학회지 52(1):97~111
Ayman Badawi / 2020 / On 1-absorbing primary ideals of commutative rings / Journal of Algebra and Its Applications 19(06):2050111
J. W. Brewer / 1981 / Power Series over Commutative Rings / Marcel Dekker, Inc
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