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Gharabagi, Zainab, Taherifar, Ali. (1404). A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS. , 13(2), 53-76. doi: 10.22044/jas.2023.13233.1729
Zainab Gharabagi; Ali Taherifar. "A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS". , 13, 2, 1404, 53-76. doi: 10.22044/jas.2023.13233.1729
Gharabagi, Zainab, Taherifar, Ali. (1404). 'A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS', , 13(2), pp. 53-76. doi: 10.22044/jas.2023.13233.1729
Gharabagi, Zainab, Taherifar, Ali. A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS. , 1404; 13(2): 53-76. doi: 10.22044/jas.2023.13233.1729

A SUBCLASS OF BAER IDEALS AND ITS APPLICATIONS

Journal of Algebraic Systems
مقاله 4، دوره 13، شماره 2، مهر 2025، صفحه 53-76    اصل مقاله (209.08 K)
نوع مقاله: Original Manuscript
شناسه دیجیتال (DOI): 10.22044/jas.2023.13233.1729
نویسندگان
Zainab Gharabagi* ؛ Ali Taherifar*
Department of Mathematics, Yasouj University, Yasouj, Iran.
چکیده
An ideal $I$ of a ring $R$ is called a right strongly Baer ideal if $r(I)=r(e)$, where $e$ is an idempotent, and there are right semicentral idempotents $e_{i}$ ($1\leq i\leq n$) with $ReR=Re_{1}R\cap Re_{2}R\cap...\cap Re_{n}R$ and each ideal $Re_{i}R$ is maximal or equals $R$. In this paper, we provide a topological characterization of this class of ideals in semiprime (resp., semiprimitive) rings. By using these results, we prove that every ideal of a ring $R$ is a right strongly Baer ideal \textit{if and only if} $R$ is a semisimple ring. Next, we give a characterization of right strongly Baer-ideals in 2-by-2 generalized triangular matrix rings, full and upper triangular matrix rings, and semiprime rings. For a semiprimitive commutative ring $R$, it is shown that $\Soc(R)$ is a right strongly Baer ideal \textit{if and only if} the set of isolated points of $\Max(R)$ is dense in it \textit{if and only if} $\Soc_{m}(R)$ is a right strongly Baer ideal. Finally, we characterize strongly Baer ideals in $C(X)$ (resp., $C(X)_{F}$).
کلیدواژه‌ها
Traingular matrix ring؛ idempotent element؛ socle of a ring؛ ring of continuous function؛ Zariski topology
مراجع
  1. M. R. Ahmadi Zand, An algebraic characterization of Blumberg spaces, Quaest. Math.,
33 (2010), 1–8.

  1. A. R. Aliabad, N. Tayarzadeh and A. Taherifar, α-Baer rings and some related concepts
via C(X), Quaest. Math., 39(3) (2016), 401–419.

  1. F. Azarpanah, Essential ideals in C(X), Period. Math. Hungar., 31 (1995), 105–112.
  2. F. Azarpanah, Intersection of essential ideals in C(X), Proc. Amer. Math. Soc., 125
(1997), 2149–2154.

  1. G. F. Birkenmeier, Idempotents and completely semiprime ideals, Comm. Algebra, 11
(1983), 567–580.

  1. G. F. Birkenmeier, M. Ghirati and A. Taherifar, When is a sum of annihilator ideals an
annihilator ideal?, Comm. Algebra, 43 (2015), 2690–2702.

  1. G. F. Birkenmeier, H. E. Heatherly, J. Y. Kim and J. K. Park, Triangular matrix repre
sentations, J. Algebra, 230 (2000), 558–595.

  1. G. F. Birkenmeier, J. K. Park and S. T. Rizvi, Generalized triangular matrix rings and the
fully invariant extending property, Rocky Mountain J. Math., 32(4) (2002), 1299–1319.

  1. W. Dietrich, On the ideal structure of C(X), Trans. Amer. Math. Soc., 152 (1970), 61–77.
  2. T. Dube and A. Taherifar, On the lattice of annihilator ideals and its applications,
Comm. Algebra, 49(6) (2021), 2444–2456.

  1. Z. Gharebaghi, M. Ghirati and A. Taherifar, On the rings of functions which are discon
tinuous on a finite set, Houston J. Math., 44(2) (2018), 721–739.

  1. M. Ghirati and A. Taherifar, Intersection of essential (resp., free) maximal ideals of
C(X), Topology Appl., 167 (2014), 62–68.

  1. L. Gillman and M. Jerison, Rings of Continuous Functions, Springer, 1976.
  2. I. Kaplansky, Rings of Operators, Benjamin, New York, 1965.
  3. T. Y. Lam, A First Course in Non-Commutative Rings, New York, springer, 1991.
  4. T. Y. Lam, Lecture on Modules and Rings, Springer, New York, 1999.
  5. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, New York, 1987.
  6. K. Samei, On the maximal spectrum of commutative semiprimitive rings, Colloq. Math.,
83(1) (2000), 5–13.

  1. G. Shin, Prime ideals and sheaf representation of a pseudo symmetric ring, Trans. Amer.
Math. Soc. Transactions, (1973), 43–60.

  1. S. A. Steinberg, Lattice-ordered ring and module, New York, Springer, 2010.
  2. A. Taherifar, A characterization of Baer-ideals, J. Algebr. Syst., 2(1) (2014), 37–51.
  3. A. Taherifar, Annihilator conditions related to the quasi-Baer condition, Hacet. J. Math.
Stat., 45(1) (2016), 95–105.

  1. A. Taherifar, Intersections of essential minimal prime ideals, Comment. Math. Univ.
Carol., 55(1) (2014), 121–130.

  1. A. Taherifar, On the socle of a commutative ring and Zariski topology, Rocky Mountain J. Math., 50(2) (2020), 707–717.
 

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