Quote:
Originally Posted by wildrabbitt
I take your point but isn't it okay to say that in the ring \(Z[\sqrt{2}]\),\(6+23\sqrt{2}\) has a factor \(2+\sqrt{2}\)?

Yes, just like in the ring Z it's right to say that 15 has a factor 3.
But if you want to divide integers in general (apart from denominator 0, of course), you get rational numbers. Similarly, if you want to divide algebraic integers in general (or for a particular ring), you get algebraic numbers (or those in your ring).