On strongly quasi $J$-ideals of commutative rings
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Let $R$ be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi $J$-ideals lying properly between the class of $J$-ideals and the class of quasi $J$-ideals. A proper ideal $I$ of $R$ is called a strongly quasi $J$-ideal if, whenever $a$, $b\in R$ and $ab\in I$, then $a^{2}\in I$ or $b\in {\rm Jac}(R)$. Firstly, we investigate some basic properties of strongly quasi $J$-ideals. Hence, we give the necessary and sufficient conditions for a ring $R$ to contain a strongly quasi $J$-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi $J$-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.
Abstract
Let $R$ be a commutative ring with identity. In this paper, we introduce a new class of ideals called the class of strongly quasi $J$-ideals lying properly between the class of $J$-ideals and the class of quasi $J$-ideals. A proper ideal $I$ of $R$ is called a strongly quasi $J$-ideal if, whenever $a$, $b\in R$ and $ab\in I$, then $a^{2}\in I$ or $b\in {\rm Jac}(R)$. Firstly, we investigate some basic properties of strongly quasi $J$-ideals. Hence, we give the necessary and sufficient conditions for a ring $R$ to contain a strongly quasi $J$-ideals. Many other results are given to disclose the relations between this new concept and others that already exist. Namely, the primary ideals, the prime ideals and the maximal ideals. Finally, we give an idea about some strongly quasi $J$-ideals of the quotient rings, the localization of rings, the polynomial rings and the trivial rings extensions.
참고문헌 (17)
- F. A. A. Almahdi, "Rings over which every semiprimary ideal is 1-absorbing primary", Comm. Algebra, 48(9), pp.3838~3845 (2020)[DOI]
- D. D. Anderson, "Ideals generated by powers of elements", Bull. Austral. Math. Soc, 49(3), pp.373~376 (1994)[DOI]
- D. D. Anderson, "Idealization of a module", J. Commut. Algebra, 1(1), pp.3~56 (2009)[DOI]
- J. T. Arnold, "On the contents of polynomials", Proc. Amer. Math. Soc, 24, pp.556~562 (1970)[DOI]
- G. C˘alug˘areanu, "UN-rings", J. Algebra Appl, 15(10), pp.9~ (2016)[DOI]
- F. DeMeyer, "Separable algebras over commutative rings" (1971), Springer
- O. Goldman, "Hilbert rings and the Hilbert Nullstellensatz", Math. Z, 54, pp.136~140 (1951)[DOI]
- : prime spectrum, Krull dimension, "J. Algebra Appl", 2005, 599~611, 10.1142[DOI]
- J. A. Huckaba, "Commutative rings with zero divisors" (1988), Marcel Dekker, Inc
- H. A. Khashan, "J-ideals of commutative rings", Int. Electron. J. Algebra, 29, pp.148~164 (2021)[DOI]
- H. A. Khashan, "Weakly J-ideals of commutative rings", Filomat, 36(2), pp.485~495 (2022)[DOI]
- H. A. Khashan, "Quasi J-ideals of commutative rings", Ricerche Mat (2022)[DOI]
- S. Ko¸c, "On strongly quasi primary ideals", Bull. Korean Math. Soc, 56(3), pp.729~743 (2019)[DOI]
- W. Krull, "Jacobsonsche Ringe, Hilbertscher Nullstellensatz, Dimensionstheorie", Math. Z, 54, pp.354~387 (1951)[DOI]
- M. Nagata, "Local rings" (1962), Interscience Publishers
- U. Tekir, "n-ideals of commutative rings", Filomat, 31(10), pp.2933~2941 (2017)[DOI]
- E. Yıldız, "(2, j)-ideals in commutative rings", C. R. Acad. Bulgare Sci, 73(9), pp.1201~1209 (2020)