References
- Atiyah, M., Macdonald, I. G. (1969). Introduction to Commutative Algebra. Reading, MA-London-Don Mills, Ontario: Addison-Wesley Publishing Co.
- Bhargava, M. (2009). Groups as unions of proper subgroups. Amer. Math. Monthly 115(5):413–422.
- Cai, M., Werner, N. (2019). Covering numbers of upper triangular matrix rings over finite fields. Involve 12(6):1005–1013.
- Coleman, E. (1996). Jónsson groups, rings and algebras. Irish Math. Soc. Bull. 36:34–45.
- Connell, G. (1963). On the group ring. Can. J. Math. 15:650–685.
- Fuchs, L. (1960). Abelian Groups. International Series of Monographs on Pure and Applied Mathematics. New York-Oxford-London-Paris: Pergamon Press.
- Garonzi, M., Kappe, L.-C., Swartz, E. (2019). On integers that are covering numbers of groups. Exp. Math. 31(2):425–443. DOI: 10.1080/10586458.2019.163426.
- Gilmer, R., Heinzer, W. (1986). On the cardinality of subrings of a commutative ring. Can. Math. Bull. 29(1):102–108.
- Kearnes, K., Oman, G. (2010). Cardinalities of residue fields of Noetherian integral domains. Commun. Algebra 38(10):3580–3588.
- Khare, A. (2009). Vector spaces as unions of proper subspaces. Linear Algebra Appl. 431(9):1681–1686.
- Kostinsky, A. (1969). Some problems for rings and lattices within the domain of general algebra. Thesis (Ph.D.) University of California, Berkeley.
- Lucchini, A., Maróti, A. (2012). Rings as the unions of proper subrings. Algebras Represent. Theory 15(6):1035–1047.
- Oman, G. (2014). Small and large ideals of an associative ring. J. Algebra Appl. 13(5):1350151.
- Peruginelli, G., Werner, N. (2018). Maximal subrings and covering numbers of finite semisimple rings. Commun. Algebra 46(11):4724–4738.
- Shelah, S. (1980). On a Problem of Kurosh, Jónsson Groups, and Applications, Word Problems II (Proc. Conf. Oxford, 1976), Amsterdam: North-Holland, pp. 373–394.
- Swartz, E., Werner, N. (2021). Covering numbers of commutative rings. J. Pure. Appl. Algebra 225(8):106622.
- Swartz, E., Werner, N. (2022). The covering numbers of rings. https://arxiv.org/pdf/2112.01667.pdf.
- Tomkinson, M. (1997). Groups as the union of proper subgroups. Math. Scand. 81(2):191–198.
- Werner, N. (2015). Covering numbers of finite rings. Amer. Math. Monthly 122(6):552–566.