## Residually small commutative rings.(English)Zbl 1400.13010

Well established in the literature is a property called residually finite which originated with groups, and since has been studied in modules and rings. A commutative ring $$R$$ is called residually finite if for every $$r \in R\setminus \{0\}=R^*$$, there exists an ideal $$I_r$$ such that $$r\notin I_r$$ and $$R/I_r$$ is finite.
This paper studies a related property of commutative rings called residually small. An infinite commutative ring $$R$$ is said to be residually small if for every non-zero $$r \in R$$, there exists and ideal $$I_r$$ such that $$r\notin I_r$$ and $$|R/I_r|< |R|$$. It is worth noting that all infinite residually finite rings are therefore also residually small with this definition. This property is related to a property for modules which is called homomorphically smaller (HS) which means for an infinite module $$M$$, $$|M/N|<|M|$$ for every non-zero submodule $$N$$ of $$M$$. Then a ring $$R$$ is HS if it is an HS module over itself.
The paper begins with an extensive preliminary section which explores many of the properties of residually small rings and especially compares these properties with other known properties in the literature. Then significant attention is turned towards studying how the residually small property behaves with respect to polynomial ring extensions, direct products, quotient rings, factor rings and integral extensions. Finally, the authors conclude by posing several interesting open questions of which we include a few. (1) If $$R$$ is a Noetherian local ring and $$I$$ is an ideal of $$R$$ with $$|R/I|=|R|$$, then $$R$$ is residually small if and only if $$R/I$$ is residually small. Does this still hold if $$R$$ is not Noetherian? (2) Suppose that $$D_1 \subseteq D_2$$ is an integral extension of domains and that $$D_1$$ is residually small and uncountable. Must $$D_2$$ be residually small? Is this question even decidable in ZFC? What if the extension were finite?

### MSC:

 13A15 Ideals and multiplicative ideal theory in commutative rings 03E10 Ordinal and cardinal numbers
Full Text:

### References:

 [1] D. Anderson and S. Chun, The set of torsion elements of a module, Comm. Alg. 42 (2014), 1835-1843. · Zbl 1290.13008 [2] —-, Zero-divisors, torsion elements, and unions of annihilators, Comm. Alg. 43 (2015), 76-83. · Zbl 1320.13013 [3] G. Baumslag, Automorphism groups of residually finite groups, J. Lond. Math. Soc. 38 (1963), 117-118. · Zbl 0124.26003 [4] K. Chew and S. Lawn, Residually finite rings, Canad. J. Math. 22 (1970), 92-101. · Zbl 0215.08901 [5] C. Faith, Note on residually finite rings, Comm. Alg. 28 (2000), 4223-4226. · Zbl 0961.16009 [6] R. Gilmer, Multiplicative ideal theory, Queen’s Papers Pure Appl. Math. 90 (1992), Queen’s University, Kingston, Ontario. [7] B. Hartley, Profinite and residually finite groups, Rocky Mountain J. Math. 7 (1977), 193-217. · Zbl 0377.20023 [8] I.D. Ion and G. Militaru, Rings with finite quotients, Bull. Math. Soc. Sci. Math. Roum. 37 (1993), 29-39. · Zbl 0855.16027 [9] I.D. Ion and C. Niţǎ, Residually finite subrings of the ring of algebraic integers, An. Univ. Buc. Mat. 56 (2007), 231-234. · Zbl 1164.11067 [10] T. Jech, Set theory, Third millennium edition, Springer Mono. Math., Springer, New York, 2002. [11] K. Kearnes and G. Oman, Cardinalities of residue fields of Noetherian integral domains, Comm. Alg. 38 (2010), 3580-3588. · Zbl 1204.13012 [12] K. Levitz and J. Mott, Rings with finite norm property, Canad. J. Math. 24 (1972), 557-565. · Zbl 0222.13006 [13] J. Lewin, Subrings of finite index in finitely generated rings, J. Algebra 5 (1967), 84-88. · Zbl 0143.05303 [14] W. Magnus, Residually finite groups, Bull. Amer. Math. Soc. 75 (1969), 305-316. · Zbl 0196.04704 [15] G. Oman, Jónsson modules over Noetherian rings, Comm. Alg. 38 (2010), 3489-3498. · Zbl 1220.13008 [16] —-, Small and large ideals of an associative ring, J. Alg. Appl. 13 (2014), 1350151. · Zbl 1298.16002 [17] G. Oman and A. Salminen, On modules whose proper homomorphic images are of smaller cardinality, Canad. Math. Bull. 2 (2012), 378-389. · Zbl 1239.13015 [18] G. Oman and R. Schwiebert, Rings which admit faithful torsion modules, Comm. Alg. 40 (2012), 2184-2198. · Zbl 1247.16002 [19] —-, Rings which admit faithful torsion modules, II, J. Alg. Appl. 11 (2012), 1250054. · Zbl 1244.16003 [20] M. Orzech and L. Ribes, Residual finiteness and the Hopf property in rings, J. Algebra 15 (1970), 81-88. · Zbl 0206.04602 [21] D. Segal, Residually finite groups, Lect. Notes Math. 1456, Springer, Berlin, 1990. · Zbl 0718.20017 [22] R. Tucci, Commutative semigroups whose proper homomorphic images are all of smaller cardinality, Kyungpook Math. J. 46 (2006), 231-233. · Zbl 1115.20053 [23] K. Varadarajan, Residual finiteness in rings and modules, J. Ramanujan Math. Soc. 8 (1993), 29-48. · Zbl 0835.16017 [24] —-, Rings with all modules residually finite, Proc. Indian Acad. Sci. Math. Sci. 109 (1999), 345-351. · Zbl 0944.16020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.