Flashcards

Definitions : Rings

Isabelle Rippon Algebraic Structures 2 2nd Year Uni English 5 cards

Cards in this deck (5)

1RingA 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 ×.
2SubringLet 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.
3Subring TestLet 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.
4Cartesian ProductIf 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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