TY - JOUR
T1 - Residually Small Commutative Rings
AU - Salminen, Adam
N1 - Let $R$ be a ring. Following the literature, $R$ is called \textit {residually finite} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $R/I_r$ is finite.
PY - 2018
Y1 - 2018
N2 - Abstract. Let R be a ring. Following the literature, R is called residually finite if for every r ∈ R\{0}, there exists an ideal Ir of R such that r /∈ Ir and R/Ir is finite. In this note, we define a strictly infinite commutative ring R with identity to be residually small if for every r ∈ R\{0}, there exists an ideal Ir of R such that r /∈ Ir and |R/Ir| < |R|. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.
AB - Abstract. Let R be a ring. Following the literature, R is called residually finite if for every r ∈ R\{0}, there exists an ideal Ir of R such that r /∈ Ir and R/Ir is finite. In this note, we define a strictly infinite commutative ring R with identity to be residually small if for every r ∈ R\{0}, there exists an ideal Ir of R such that r /∈ Ir and |R/Ir| < |R|. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.
UR - https://projecteuclid.org/journals/journal-of-commutative-algebra/volume-10/issue-2/Residually-small-commutative-rings/10.1216/JCA-2018-10-2-187.short
U2 - 10.1216/JCA-2018-10-2-187
DO - 10.1216/JCA-2018-10-2-187
M3 - Article
VL - 10
JO - Journal of Commutative Algebra
JF - Journal of Commutative Algebra
ER -