Ring (English)
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Definition (undergraduate level)
A set, with two binary operations. The operations are usually called addition and multiplication, and the set must form a commutative group under the addition operation. The set must also be closed under multiplication, and multiplication must be associative, and distributive over addition.
Definition (undergraduate level)
A commutative ring with unit is a set R with binary operations + : R2 → R and . : R2 → R and distinguished elements 0,1 ∈ R such that
(R,+) is a commutative group with identity 0
. is commutative, associative and 1.x=x ∀x ∈ R
(x+y).z=(x.z)+(y.z) is identically true in R
See also domain, unit, field.
(R,+) is a commutative group with identity 0
. is commutative, associative and 1.x=x ∀x ∈ R
(x+y).z=(x.z)+(y.z) is identically true in R
See also domain, unit, field.
Relations
- broader:
- (en) Algebraic object
- narrower:
- (en) Division ring or skew field
- (en) Eisenstein integers
- (en) Euclidean ring
- (en) Gaussian integers
- (en) Integral domain
- (en) Polynomial ring
- (en) Quotient ring
- (en) Subring
- references:
- (en) Binary operation
- (en) Group
- (en) Set
- referenced:
- (en) Area of a ring
- (en) Module
- see also:
- (en) Domain
- (en) Field
- (en) Ring or annulus
- (en) Unit
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