Chickenmancer

$$\frac1{x_1} + \frac1{x_2} + \frac1{x_3}+ \frac1{x_4} = \frac{x_2x_3x_4 + x_1x_3x_4 + x_1x_2x_4+x_1x_2x_3}{x_1x_2x_3x_4}=\frac{6}{7}$$ Since $x_i$ are all natural numbers, you can look to natural number solutions to $$7x_2x_3x_4 + 7x_1x_3x_4 + 7x_1x_2x_4+7x_1x_2x_3 - 6 x_1x_2x_3x_4=0$$

There are many reasons to study abelian groups, one reason you may wish to study them, or study their properties, is to understand the decomposition of more sophisticated objects, which have an underlying abelian group structure, eg vector spaces, modules, fields, division algebras, and the list goes on and on and on . . .

@samjoe, increasing is different than strictly increasing.

Yes can imagine that dropping the domain condition would break many of the theorems which classify ideals of products.

I don't believe there are generalizations of this. For example, the (direct) product of two integral domains is never an integral domain: (1,0)*(0,1)=(0,0). So the product of two PID's will never be a domain, and therefore never be a PID.

What theorems in particular about products of PID/UFD?