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''left''

A ring (usually assumed to be associative and with a unit element) in which the left ideals form a chain. In other words, $R$ is a left chain ring if $R$ is a left chain module over itself. Every left chain ring is local. Right chain rings are defined similarly. Commutative chain rings are often called normed rings. See also [[Discretely-normed ring|Discretely-normed ring]].