On the arithmetic of stable domains

References

• Anderson, D. D., Anderson, D. F., Zafrullah, M. (1992). Atomic domains in which almost all atoms are prime. Commun. Algebra 20(5):1447–1462. DOI: 10.1080/00927879208824413.
   Web of Science ®Google Scholar
• Anderson, D. D., Huckaba, J. A., Papick, I. J. (1987). A note on stable domains. Houston J. Math. 13(1):13–17.
   Web of Science ®Google Scholar
• Anderson, D. D., Mott, J., Zafrullah, M. (1992). Finite character representations for integral domains. Boll. Unione Mat. Ital. 6:613–630.
   Google Scholar
• Anderson, D. D., Mott, J. L. (1992). Cohen-Kaplansky domains: Integral domains with a finite number of irreducible elements. J. Algebra 148(1):17–41. DOI: 10.1016/0021-8693(92)90234-D.
   Web of Science ®Google Scholar
• Baeth, N. R., Smertnig, D. Lattices over Bass rings and graph agglomerations. Algebr. Represent. Theory DOI: 10.1007/s10468-021-10040-2.
   Web of Science ®Google Scholar
• Bass, H. (1963). On the ubiquity of Gorenstein rings. Math. Z. 82(1):8–28. DOI: 10.1007/BF01112819.
   Google Scholar
• Bazzoni, S. (2000). Divisorial domains. Forum Math. 12:397–419.
   Web of Science ®Google Scholar
• Bazzoni, S., Salce, L. (1996). Warfield domains. J. Algebra 185(3):836–868. DOI: 10.1006/jabr.1996.0353.
   Web of Science ®Google Scholar
• Brantner, J., Geroldinger, A., Reinhart, A. (2020). On monoids of ideals of orders in quadratic number fields. In: Facchini, A., Fontana, M., Geroldinger, A., Olberding, B. Advances in rings, modules, and factorizations, Vol. 321. Switzerland: Springer, pp. 11–54.
   Google Scholar
• Dobbs, D. E., Fontana, M. (1987). Seminormal rings generated by algebraic integers. Mathematika 34(2):141–154. DOI: 10.1112/S0025579300013395.
   Web of Science ®Google Scholar
• Fan, Y., Geroldinger, A., Kainrath, F., Tringali, S. (2017). Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules. J. Algebra Appl. 16(12):1750234–1750242. DOI: 10.1142/S0219498817502346.
   Web of Science ®Google Scholar
• Gabelli, S. (2006). Generalized Dedekind domains. In: Brewer, J. W., Glaz, S., Heinzer, W., Olberding, B. Multiplicative Ideal Theory in Commutative Algebra. New York, NY: Springer, pp. 189–206.
   Google Scholar
• Gabelli, S. (2014). Ten problems on stability of domains. In: Fontana, M., Frisch, S., Glaz, S. Commutative Algebra. New York, NY: Springer, pp. 175–193.
   Google Scholar
• Gabelli, S., Roitman, M. (2019). On finitely stable domains, I. J. Commut. Algebra 11(1):49–67. DOI: 10.1216/JCA-2019-11-1-49.
   Web of Science ®Google Scholar
• Gabelli, S., Roitman, M. (2020). On finitely stable domains II. J. Commut. Algebra 12:179–198.
   Web of Science ®Google Scholar
• Gao, W., Liu, C., Tringali, S., Zhong, Q. (2021). On half-factoriality of transfer Krull monoids. Commun. Algebra 49(1):409–420. DOI: 10.1080/00927872.2020.1800720.
   Web of Science ®Google Scholar
• García-Sánchez, P. A., O’Neill, C., Webb, G. (2019). The computation of factorization invariants for affine semigroups. J. Algebra Appl. 18(1):1950019–1950021. DOI: 10.1142/S0219498819500191.
   Web of Science ®Google Scholar
• Geroldinger, A. (2016). Sets of lengths. Amer. Math. Monthly 123:960–988.
   Web of Science ®Google Scholar
• Geroldinger, A., Gotti, F., Tringali, S. (2021). On strongly primary monoids, with a focus on Puiseux monoids. J. Algebra 567:310–345. DOI: 10.1016/j.jalgebra.2020.09.019.
   Web of Science ®Google Scholar
• Geroldinger, A., Grynkiewicz, D. J., Schmid, W. A. (2011). The catenary degree of Krull monoids I. J. Théor. Nombres Bordeaux. 23(1):137–169. DOI: 10.5802/jtnb.754.
   Google Scholar
• Geroldinger, A., Halter-Koch, F. (2006). Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics. Vol. 278. Boca Raton, FL: Chapman & Hall/CRC Press.
   Google Scholar
• Geroldinger, A., Halter-Koch, F., Hassler, W., Kainrath, F. (2003). Finitary monoids. Semigroup Forum 67(1):1–21. DOI: 10.1007/s002330010141.
   Web of Science ®Google Scholar
• Geroldinger, A., Hassler, W., Lettl, G. (2007). On the arithmetic of strongly primary monoids. Semigroup Forum 75(3):567–587. DOI: 10.1007/s00233-007-0721-y.
   Web of Science ®Google Scholar
• Geroldinger, A., Kainrath, F., Reinhart, A. (2015). Arithmetic of seminormal weakly Krull monoids and domains. J. Algebra 444:201–245. DOI: 10.1016/j.jalgebra.2015.07.026.
   Web of Science ®Google Scholar
• Geroldinger, A., Reinhart, A. (2019). The monotone catenary degree of monoids of ideals. Int. J. Algebra Comput. 29(3):419–457. DOI: 10.1142/S0218196719500097.
   Web of Science ®Google Scholar
• Geroldinger, A., Roitman, M. (2020). On strongly primary monoids and domains. Commun. Algebra 48(9):4085–4099. DOI: 10.1080/00927872.2020.1755678.
   PubMed Web of Science ®Google Scholar
• Geroldinger, A., Zhong, Q. (2018). Long sets of lengths with maximal elasticity. Can. J. Math. 70(6):1284–1318. DOI: 10.4153/CJM-2017-043-4.
   Web of Science ®Google Scholar
• Geroldinger, A., Zhong, Q. (2020). Factorization theory in commutative monoids. Semigroup Forum 100:22–51.
   Web of Science ®Google Scholar
• Goeters, H. P. (1998). Noetherian stable domains J. Algebra 202(1):343–356. DOI: 10.1006/jabr.1997.7305.
   Web of Science ®Google Scholar
• Greither, C. (1982). On the two generator problem for the ideals of a one-dimensional ring. J. Pure Appl. Algebra 24(3):265–276. DOI: 10.1016/0022-4049(82)90044-5.
   Web of Science ®Google Scholar
• Halter-Koch, F. (1998). Ideal Systems. An Introduction to Multiplicative Ideal Theory. New York, NY: Marcel Dekker.
   Google Scholar
• Halter-Koch, F. (2011). Multiplicative ideal theory in the context of commutative monoids. In: Fontana, M., Kabbaj, S.-E., Olberding, B., Swanson, I. Commutative Algebra: Noetherian and Non-Noetherian Perspectives. New York, NY: Springer, pp. 203–231.
   Google Scholar
• Halter-Koch, F. (2013). Quadratic Irrationals, Pure and Applied Mathematics. Vol. 306. Boca Raton, FL: Chapman & Hall/CRC Press.
   Google Scholar
• Halter-Koch, F. (2020). An Invitation to Algebraic Numbers and Algebraic Functions. Pure and Applied Mathematics. Boca Raton, FL: CRC Press.
   Google Scholar
• Hassler, W. (2009). Properties of factorizations with successive lengths in one-dimensional local domains. J. Commut. Algebra 1(2):237–268. DOI: 10.1216/JCA-2009-1-2-237.
   Web of Science ®Google Scholar
• Lipman, J. (1971). Stable ideals and Arf rings. Amer. J. Math. 93(3):649–685. DOI: 10.2307/2373463.
   Web of Science ®Google Scholar
• Loper, K. A. (2006). Almost Dedekind Domains Which Are Not Dedekind. In: Brewer, J. W., Glaz, S., Heinzer, W., Olberding, B. Multiplicative Ideal Theory in Commutative Algebra. Springer, pp. 279–292.
   Google Scholar
• Matlis, E. (1968). Reflexive domains. J. Algebra. 8(1):1–33. DOI: 10.1016/0021-8693(68)90031-8.
   Web of Science ®Google Scholar
• Matlis, E. (1970). The two-generator problem for ideals. Michigan Math. J. 17:257–265.
   Web of Science ®Google Scholar
• Olberding, B. (1998). Globalizing local properties of Prüfer domains. J. Algebra 205(2):480–504. DOI: 10.1006/jabr.1997.7406.
   Web of Science ®Google Scholar
• Olberding, B. (2001). On the classification of stable domains. J. Algebra 243:177–197.
   Web of Science ®Google Scholar
• Olberding, B. (2001a). Stability, duality, 2-generated ideals and a canonical decomposition of modules. Rend. Sem. Mat. Univ. Padova 106:261–290.
   Google Scholar
• Olberding, B. (2001b). Stability of ideals and its applications. In: Anderson, D. D., Papick, I. J., eds. Ideal Theoretic Methods in Commutative Algebra. Lecture Notes in Pure and Applied Mathematics (D.D. Anderson and I.J. Papick, eds.), Vol. 220. New York, NY: Marcel Dekker, pp. 319–341.
   Google Scholar
• Olberding, B. (2002). On the structure of stable domains. Commun. Algebra 30:877–895.
   Web of Science ®Google Scholar
• Olberding, B. (2014). Finitely Stable Rings. Commutative Algebra. New York: Springer, pp. 269–291.
   Google Scholar
• Olberding, B. (2014). One-dimensional bad Noetherian domains. Trans. Amer. Math. Soc. 366(8):4067–4095.
   Web of Science ®Google Scholar
• Olberding, B. (2016). One-dimensional stable rings. J. Algebra 456:93–122.
   Web of Science ®Google Scholar
• Olberding, B., Reinhart, A. (2020). Radical factorization in finitary ideal systems. Commun. Algebra 48(1):228–253. DOI: 10.1080/00927872.2019.1640237.
   PubMed Web of Science ®Google Scholar
• Omidali, M. (2012). The catenary and tame degree of numerical monoids generated by generalized arithmetic sequences. Forum Math. 24:627–640.
   Web of Science ®Google Scholar
• O’Neill, C., Pelayo, R. (2017). Factorization invariants in numerical monoids. In: Harrington, H. A., Omar, M., Wright, M., eds. Algebraic and Geometric Methods in Discrete Mathematics, Contemporary Mathematics, Vol. 685. Providence, RI: American Mathematical Society, pp. 231–249.
   Google Scholar
• O’Neill, C., Pelayo, R. (2018). Realisable sets of catenary degrees of numerical monoids. Bull. Australian Math. Soc. 97:240–245.
   Web of Science ®Google Scholar
• Philipp, A. (2015). A characterization of arithmetical invariants by the monoid of relations II: The monotone catenary degree and applications to semigroup rings. Semigroup Forum 90:220–250.
   Web of Science ®Google Scholar
• Querré, J. (1976). Intersections d’anneaux intègres. J. Algebra 43(1):55–60.
   Web of Science ®Google Scholar
• Sally, J. D., Vasconcelos, W. V. (1973). Stable rings and a problem of Bass. Bull. Amer. Math. Soc. 79(3):574–576. DOI: 10.1090/S0002-9904-1973-13209-4.
   Web of Science ®Google Scholar
• Tringali, S. (2019). Structural properties of subadditive families with applications to factorization theory. Isr. J. Math. 234(1):1–35. DOI: 10.1007/s11856-019-1922-2.
   Web of Science ®Google Scholar