I am currently reading John B Fraleigh's book A First Course in Abstract Algebra. The author in chapter 23, insists on the point that it is necessary for one to show that the R is closed under multiplication (along with the other Ring axioms) in order to completely show that is a Ring. However, in the definition of a Ring, I've come across only three axioms, which are as follows:
- (R,+) must be an Abelian Group
- must be associative
- For all a,b,c $\in$R the left distributive law and the right distributive law holds.
Now where is it mentioned that a*b must belong to R for all a,b $\in$ R. There are consequences of this fact, namely in showing that the set of all pure imaginary complex numbers ri for r $\in$ $\Re$ (with the usual addition and multiplication) is not a Ring. Since any two pure imaginary numbers are not closed under multiplication, the author writes that this set does not define a Ring.