A closer look at primal and pseudo-irreducible ideals with applications to rings of functions

References

• Afrooz, S., Azarpanah, F., Karamzadeh, O. A. S. (2015). Goldie dimension of rings of fractions of C(X). Quaest. Math. 38(1):139–154. DOI: 10.2989/16073606.2014.923189.
   Web of Science ®Google Scholar
• Al-Ezeh, H. (1989). Exchange PF-rings and almost PP-rings. Int. J. Math. Math. Sci. 12(4):725–727. DOI: 10.1155/S016117128900089X.
   Google Scholar
• Al-Ezeh, H. (1989). Pure ideals in commutative reduced Gelfand rings with unity. Arch. Math. (Basel) 53(3):266–269. DOI: 10.1007/BF01277063.
   Web of Science ®Google Scholar
• Almahdi, F. A. A., Tamekkante, M., Mamouni, A. (2020). Rings over which every semi-primary ideal is 1-absorbing primary. Commun. Algebra 48(9):3838–3845. DOI: 10.1080/00927872.2020.1749645.
   Web of Science ®Google Scholar
• Artico, G., Marconi, U. (1976). On compactness of minimal spectrum. Rend. Sem. Mat. Univ. Padova 56:79–84.
   Google Scholar
• Azarpanah, F. (1999/2000). Algebraic properties of some compact spaces. Real Anal. Exch. 25(1):317–328. DOI: 10.2307/44153077.
   Google Scholar
• Azarpanah, F. (1995). Essential ideals in C(X). Period. Math. Hungar. 31(2):105–112. DOI: 10.1007/BF01876485.
   Google Scholar
• Azarpanah, F. (2002). When is C(X) a clean ring? Acta Math. Hungar. 94(1–2):53–58. DOI: 10.1023/A:1015654520481.
   Web of Science ®Google Scholar
• Azarpanah, F., Esmaeilvandi, D., Salehi, A. R. (2019). Depth of ideals of C(X). J. Algebra 528:474–496. DOI: 10.1016/j.jalgebra.2019.03.018.
   Web of Science ®Google Scholar
• Azarpanah, F., Farokhpay, F., Ghashghaei, E. (2019). Annihilator-stability and unique generation in C(X). J. Algebra Appl. 18(7):1950122. DOI: 10.1142/S0219498819501226.
   Web of Science ®Google Scholar
• Azarpanah, F., Ghashghaei, E., Ghoulipour, M. (2020). C(X): Something old and something new. Commun. Algebra 49(1):185–206. DOI: 10.1080/00927872.2020.1797070.
   Web of Science ®Google Scholar
• Azarpanah, F., Karamzadeh, O. A. S., Rezai Aliabad, A. (1999). On z°-ideals in C(X). Fund. Math. 160(1):15–25. https://eudml.org/doc/212377.
   Web of Science ®Google Scholar
• Azarpanah, F., Olfati, A. R. (2015). On ideals of ideals in C(X). Bull. Iran. Math. Soc. 41(1):23–41.
   Web of Science ®Google Scholar
• Azarpanah, F., Salehi, A. R. (2016). Ideal structure of the classical ring of quotients of C(X). Topol. Appl. 209:170–180. DOI: 10.1016/j.topol.2016.06.008.
   Web of Science ®Google Scholar
• Banaschewski, B. (1961). On the components of ideals in commutative rings. Arch. Math. (Basel) 12:22–29. DOI: 10.1007/BF01650518.
   Google Scholar
• Banerjee, B., Ghosh, S. K., Henriksen, M. (2009). Unions of minimal prime ideals in rings of continuous functions on compact spaces. Algebra Universalis 62(2–3):239–246. DOI: 10.1007/s00012-010-0051-x.
   Web of Science ®Google Scholar
• Borceux, F., Van den Bossche, G. (1983). Algebra in a Localic Topos with Applications to Ring Theory. Lecture Notes in Mathematics. Berlin: Springer-Verlag.
   Google Scholar
• Burgess, W. D., Raphael, R. (2020). Tychonoff spaces and a ring theoretic order on C(X). Topol. Appl. 279:107250. DOI: 10.1016/j.topol.2020.107250.
   Web of Science ®Google Scholar
• Contessa, M. (1982). On pm-rings. Commun. Algebra 10(1):93–108. DOI: 10.1080/00927878208822703.
   Web of Science ®Google Scholar
• Cornish, W. H. (1970). Abelian Rickart semirings. Thesis, School of Mathematical Sciences, The Flinders University of South Australia.
   Google Scholar
• Dashiell, F., Hager, A., Henriksen, M. (1980). Order-Cauchy completions of rings and vector lattices of continuous functions. Can. J. Math. 32(3):657–685. DOI: 10.4153/CJM-1980-052-0.
   Web of Science ®Google Scholar
• Dauns, J., Fuchs, L. (2004). Torsion-freeness for rings with zero-divisors. J. Algebra Appl. 03(03):221–237. DOI: 10.1142/S0219498804000885.
   Google Scholar
• De Marco, G., Orsatti, A. (1971). Commutative rings in which every prime ideal is contained in a unique maximal ideal. Proc. Amer. Math. Soc. 30:459–466. DOI: 10.2307/2037716.
   Web of Science ®Google Scholar
• Dietrich, W. (1970). On the ideal structure of C(X). Trans. Amer. Math. Soc. 152:61–77. DOI: 10.2307/1995638.
   Google Scholar
• Estaji, A. A., Karamzadeh, O. A. S. (2003). On C(X) modulo its socle. Commun. Algebra 31(4):1561–1571. DOI: 10.1081/AGB-120018497.
   Web of Science ®Google Scholar
• Fuchs, L. (1948). A condition under which an irreducible ideal is primary. Quart. J. Math. Oxford Ser. 19:235–237.os-19.1.235. DOI: 10.1093/qmath/.
   Google Scholar
• Fuchs, L. (1950). On primal ideals. Proc. Amer. Math. Soc. 1:1–6. DOI: 10.2307/2032421.
   Web of Science ®Google Scholar
• Fuchs, L. (1947). On quasi-primary ideals. Acta Univ. Szeged. Sect. Sci. Math. 11:174–183.
   Google Scholar
• Fuchs, L., Heinzer, W., Olberding, B. (2005). Commutative ideal theory without finiteness conditions: primal ideals. Trans. Amer. Math. Soc. 357(7):2771–2798. DOI: 10.1090/S0002-9947-04-03583-4.
   Web of Science ®Google Scholar
• Ghashghaei, E. (2022). Variations of essentiality of ideals in commutative rings. J. Algebra Appl. 21(3):2250056. DOI: 10.1142/S0219498822500566.
   Google Scholar
• Gillman, L., Henriksen, M. (1956). Rings of continuous functions in which every finitely generated ideal is principal. Trans. Amer. Math. Soc. 82(2):366–391. DOI: 10.2307/1993054.
   Google Scholar
• Gillman, L., Jerison, M. (1960). Rings of Continuous Functions. The University Series in Higher Mathematics. Princeton, NJ: Van Nostrand.
   Google Scholar
• Gillman, L., Kohls, C. W. (1959/1960). Convex and pseudoprime ideals in rings of continuous functions. Math. Z. 72:399–409. DOI: 10.1007/BF01162963.
   Google Scholar
• Gilmer, R. W. (1962). Rings in which semi-primary ideals are primary. Pacific J. Math. 12:1273–1276. DOI: 10.2140/pjm.1962.12.1273.
   Web of Science ®Google Scholar
• Glaz, S. (1989). Commutative Coherent Rings. Lecture Notes in Mathematics. Berlin, Germany: Springer-Verlag.
   Google Scholar
• Hedayat, S., Rostami, E. (2018). A characterization of commutative rings whose maximal ideal spectrum is Noetherian. J. Algebra Appl. 17(1):1850003. DOI: 10.1142/S0219498818500032.
   Web of Science ®Google Scholar
• Hedayat, S., Rostami, E. (2017). Decompositions of ideals into pseudo-irreducible ideals. Commun. Algebra 45(4):1711–1718. DOI: 10.1080/00927872.2016.1222410.
   Web of Science ®Google Scholar
• Henriksen, M., Wilson, R. (1992). When is C(X)/P a valuation ring for every prime ideal P?. Topol. Appl. 44(1–3): 175–180. DOI: 10.1016/0166-8641(92)90091-D.
   Google Scholar
• Iroz, J., Rush, D. (1984). Associated prime ideals in non-Noetherian rings. Can. J. Math. 36(2):344–360. DOI: 10.4153/CJM-1984-021-6.
   Web of Science ®Google Scholar
• Jensen, C. U. (1964). A remark on arithmetical rings. Proc. Amer. Math. Soc. 15(6):951–954. DOI: 10.2307/2034915.
   Google Scholar
• Juett, J. (2012). Generalized comaximal factorization of ideals. J. Algebra 352:141–166. DOI: 10.1016/j.jalgebra.2011.11.008.
   Web of Science ®Google Scholar
• Karamzadeh, O. A. S., Rostami, M. (1985). On the intrinsic topology and some related ideals of C(X). Proc. Amer. Math. Soc. 93(1):179–184. DOI: 10.2307/2044578.
   Web of Science ®Google Scholar
• Kist, J. (1974). Two characterizations of commutative Baer rings. Pacific J. Math. 50(1):125–134. https://msp.org/pjm/1974/50-1/p14.xhtml.
   Web of Science ®Google Scholar
• Klingler, L., McGovern, W. W. (2019). Local types of classical rings. In: Badawi, A., Coykendall, J., eds. Advances in Commutative Algebra. Trends in Mathematics, Singapore: Birkhäuser/Springer, pp. 159–170.
   Google Scholar
• Knox, M. L., Levy, R., McGovern, W. W., Shapiro, J. (2009). Generalizations of complemented rings with applications to rings of functions. J. Algebra Appl. 08(01):17–40. DOI: 10.1142/S0219498809003138.
   Google Scholar
• Krull, W. (1929). Idealtheorie in Ringen ohne Endlichkeitsbedingung. Math. Ann. 101(1):729–744. DOI: 10.1007/BF01454872.
   Google Scholar
• Lam, T. Y. (2001). A First Course in Noncommutative Rings, Graduate Texts in Mathematics, Vol. 131, 2nd ed. New York: Springer-Verlag.
   Google Scholar
• Lam, T. Y. (1999). Lectures on Modules and Rings. Graduate Texts in Mathematics, Vol. 189. New York: Springer-Verlag.
   Google Scholar
• McAdam, S., Swan, R. G. (2004). Unique comaximal factorization. J. Algebra 276(1):180–192. DOI: 10.1016/j.jalgebra.2004.02.007.
   Web of Science ®Google Scholar
• Martinez, J., Woodward, S. (1992). Bezout and Prüfer f-rings. Commun. Algebra 20(10):2975–2989. DOI: 10.1080/00927879208824500.
   Web of Science ®Google Scholar
• Matlis, E. (1983). The minimal prime spectrum of a reduced ring. Illinois J. Math. 27(3):353–391. DOI: 10.1215/ijm/1256046365.
   Web of Science ®Google Scholar
• McGovern, W. W. (2006). Neat rings. J. Pure Appl. Algebra 205(2):243–265. DOI: 10.1016/j.jpaa.2005.07.012.
   Web of Science ®Google Scholar
• Nicholson, W. K. (1977). Lifting idempotents and exchange rings. Trans. Amer. Math. Soc. 229:269–278. DOI: 10.2307/1998510.
   Web of Science ®Google Scholar
• Niefeld, S., Rosenthal, K. (1987). Sheaves of integral domains on stone spaces. J. Pure Appl. Algebra 47(2):173–179. DOI: 10.1016/0022-4049(87)90060-0.
   Web of Science ®Google Scholar
• Noether, E. (1921). Idealtheorie in Ringbereichen. Math. Ann. 83(1–2):24–66. DOI: 10.1007/BF01464225.
   Google Scholar
• Rowen, L. H. (1991). Ring Theory, Student ed. San Diego: Academic Press.
   Google Scholar
• Taherifar, A. (2014). Intersections of essential minimal prime ideals. Comment. Math. Univ. Carolin. 55(1):121–130. https://dml.cz/handle/10338.dmlcz/143572.
   Google Scholar
• Lohar.com. Available at https://lohar.com/mithelpdesk/hd1103.pdf (accessed 6.5.2022).
   Google Scholar
• Lohar.com. Available at https://lohar.com/mithelpdesk/hd1201.pdf (accessed 6.5.2022)
   Google Scholar