About Group, Ring and Field

Group:

G1) Closure under addition [if a and b belong to S, then a+b is also in S]

G2) Associativity of addition [a+(b+c) = (a+b)+c for all a, b, c in S]

G3) Additive identity [There is an element 0 in R such that a+0=0+a=a for all a in S]

G4) Additive inverse [For each a in S there is an element -a in S such that a+(-a)=(-a)+a=0]

Abelian Group:

AG) Commutativity of addition [a+b=b+a for all a, b in S]

Ring:

R1) Closure under multiplication [If a and b belong to S, then ab is also in S]

R2) Associativity of multiplication [a(bc)=(ab)c for all a, b, c in S]

R3) Distributive laws [a(b+c)=ab+ac for all a, b, c in S (a+b)c=ac+bc for all a,b, c in S]

Commutative Ring:

CR) Commutativity of multiplication [ab=ba for all a, b in S]

Integral Domain:

ID1) Multiplicative identity [There is an element 1 in S sucht that a1=1a=a for all a in S]

ID2) No zero divisors [If a, b in S and ab = 0, then either a=0 or b=0]

Field:

F) Multiplicative invers [if a belongs to S and a != 0, there is an element 1/a in S such that (a)(1/a)=(1/a)(a)=1]

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