abstract algebra

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Can you name the axioms which define the following objects?

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Group
GxG->G 
A*(B*C) = (A*B)*C 
for all g in G, g^-1 is in G 
for some e and all g in G, e*g = g = g*e 
Abelian Group
A*B = B*A 
Vector Space
V + V -> V 
a * V -> V 
(V, +) 
S 
Ring
0 + r = r for some 0 and any r in R 
1 * r = r for some 1 and any r in R 
a * (b + c) = a*b + a*c 
for all r in R, -r is also in R 
Field
for all f in {F-0}, 1/f is in F 

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Created Jul 4, 2012ReportNominate
Tags:math, field, group, ring