Fields {ring, field} can have only addition, subtraction, and multiplication, with no division. Rings have distributive property. Rings can have unit element or not. In rings, one operation is commutative, and other operation is closed, associative, and not necessarily commutative. If rings have unit element and inverse elements, they are non-commutative {division ring}. Rings have elements {center, ring} that do commute. Ring elements {kernel, ring} can correspond to another ring's zero elements, by homomorphism.
Rings can be associative and non-associative algebras {Lie algebra}.
Commutative rings {Noetherian ring} can have ideals in forms (Amalie Noether or Emmy Noether).
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Date Modified: 2022.0225