Definitions : Rings

Question	Answer
Ring	A ring is a set R and two binary operations, written + and ×, on R which satisfies the following conditions: (R1) (R, +) is an abelian group with identity 0. (R2) × is associative. (R3) × is distributive over +: ie. a × (b + c) = (a × b) + (a × c) and (b + c) × a = (b × a) + (c × a) for all a, b, c ∈ R. (R4) there exists an element 1 ∈ R, different from 0, that is an identity for ×.
Subring	Let R be a ring and S ⊆ R. Then S is a subring of R if it is a ring in its own right with respect to the same addition and multiplication as in R and S contains 1R.
Subring Test	Let R be a ring and S ⊆ R. Then S is a subring of R, iff: (i) 1R ∈ S; (ii) r + s, r × s ∈ S, for all r, s ∈ S; (iii) −r ∈ S for all r ∈ S.
Cartesian Product	If R1 and R2 are rings then the Cartesian product R1 × R2 with operations + and × defined by (r1, r2) + (s1, s2) = (r1 + s1, r2 + s2) and (r1, r2) × (s1, s2) = (r1 × s1, r2 × s2) is a ring

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	Created by Isabelle Rippon over 1 year ago