Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
semi $n$-ideal; semiprime ideal; $n$-ideal
semi $n$-ideal; semiprime ideal; $n$-ideal
Summary:
Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$-ideal of $R$ if for $a,b\in R$, $ab\in I$ and $a\notin \sqrt {0}$ imply $b\in I$. We give a new generalization of the concept of $n$-ideals by defining a proper ideal $I$ of $R$ to be a semi $n$-ideal if whenever $a\in R$ is such that $a^{2}\in I$, then $a\in \sqrt {0}$ or $a\in I$. We give some examples of semi \hbox {$n$-ideal} and investigate semi $n$-ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new class of ideals. Moreover, we prove that every proper ideal of a zero dimensional general ZPI-ring $R$ is a semi $n$-ideal if and only if $R$ is a UN-ring or $R\cong F_{1}\times F_{2}\times \cdots \times F_{k}$, where $F_{i}$ is a field for $i=1,\dots ,k$. Finally, for a ring homomorphism $f\colon R\rightarrow S$ and an ideal $J$ of $S$, we study some forms of a semi $n$-ideal of the amalgamation $R\bowtie ^{f}J$ of $R$ with $S$ along $J$ with respect to $f$.
References:
[1] Badawi, A.: On weakly semiprime ideals of commutative rings. Beitr. Algebra Geom. 57 (2016), 589-597. DOI 10.1007/s13366-016-0283-9 | MR 3535069 | Zbl 1349.13005
[2] Călugăreanu, G.: $UN$-rings. J. Algebra Appl. 15 (2016), Article ID 1650182, 9 pages. DOI 10.1142/S0219498816501826 | MR 3575972 | Zbl 1397.16037
[3] Çelikel, E. Yetkin: Generalizations of $n$-ideals of commutative rings. Erzincan Univ. J. Sci. Technol. 12 (2019), 650-657. DOI 10.18185/erzifbed.471609 | MR 4355689
[4] D'Anna, M., Finocchiaro, C. A., Fontana, M.: Properties of chains of prime ideals in an amalgamated algebra along an ideal. J. Pure Appl. Algebra 214 (2010), 1633-1641. DOI 10.1016/j.jpaa.2009.12.008 | MR 2593689 | Zbl 1191.13006
[5] D'Anna, M., Fontana, M.: An amalgamated duplication of a ring along an ideal: The basic properties. J. Algebra Appl. 6 (2007), 443-459. DOI 10.1142/S0219498807002326 | MR 2337762 | Zbl 1126.13002
[6] Gilmer, R.: Multiplicative Ideal Theory. Queen's Papers in Pure and Appled Mathematics 90. Queen's University, Kingston (1992). MR 1204267 | Zbl 0804.13001
[7] Khashan, H. A., Bani-Ata, A. B.: $J$-ideals of commutative rings. Int. Electron. J. Algebra 29 (2021), 148-164. DOI 10.24330/ieja.852139 | MR 4206318 | Zbl 1467.13005
[8] Khashan, H. A., Çelikel, E. Yetkin: Quasi $J$-ideals of commutative rings. (to appear) in Ric. Mat (2022). DOI 10.1007/s11587-022-00716-2 | MR 4394285
[9] Khashan, H. A., Çelikel, E. Yetkin: Weakly $J$-ideals of commutative rings. Filomat 36 (2022), 485-495. DOI 10.2298/FIL2202485K | MR 4394285
[10] Tekir, U., Koc, S., Oral, K. H.: $n$-ideals of commutative rings. Filomat 31 (2017), 2933-2941. DOI 10.2298/FIL1710933T | MR 3639382 | Zbl 07418085
Search
Partner of
