Geodesic
On the Cardinality of Subrings of a Commutative Ring
Canadian mathematical bulletin, Tome 29 (1986) no. 1, pp. 102-108

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If R is an uncountable commutative ring, it is shown that there exists a proper subring of R having the same cardinality as R. It is also shown that if |R| = ω is an uncountable regular cardinal, and if R1 is a subring of R containing an identity of R and such that |R1| < ω, then there exists a proper R1 -subalgebra S of R such that |S| = |R|.
DOI : 10.4153/CMB-1986-019-0
Mots-clés : 13B02, 13C05, 13E05 
Gilmer, Robert; Heinzer, William. On the Cardinality of Subrings of a Commutative Ring. Canadian mathematical bulletin, Tome 29 (1986) no. 1, pp. 102-108. doi: 10.4153/CMB-1986-019-0
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     title = {On the {Cardinality} of {Subrings} of a {Commutative} {Ring}},
     journal = {Canadian mathematical bulletin},
     pages = {102--108},
     year = {1986},
     volume = {29},
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     doi = {10.4153/CMB-1986-019-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1986-019-0/}
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