How to define the '*' operation in the ring How to define the '*'(multiplication) operation in the ring, where the set is real positive numbers and the '+' operation is multiplication, so that the ring will be without zero divisors? The same question for the ring, where the set is rational positive numbers.
I understand that the zero element in this ring is '1', so we need the equation a*b=1 to be unsolvable if a and b are not 1. But I've tried all the typical operations such as addition or exponentiation and haven't succeeded.
 A: Let's look at the abelian group structure of your set $\mathbb R _+$ - it is isomorphic to $\mathbb R$ with usual addition, via 
$$f : \mathbb R \rightarrow \mathbb R _+$$
$$f(x) = e^x$$
since
$$e^{x+y} = e^x \cdot e^y$$
Now, just transfer the usual multiplication of $\mathbb R$ using this isomorphism, that is, define multiplication $*$ on $\mathbb R_+$ as
$$e^x * e^y = e^{xy}$$
which is 
$$x * y = e^{\ln x \ln y}$$

The case of positive rationals is a bit subtler. Again, look at the additive structure of our to-be ring. Every positive rational can be uniquely represented as $p_1^{n_1} p_2^{n_2} \dots p_k ^{n_k}$, where $p_i$ are primes, and $n_i \in \mathbb Z$. Multiplication of rationals corresponds to adding the $n_i$'s, which provides an isomorphism of abelian groups
$$\mathbb Q_+ \cong \bigoplus\limits_{i=0}^{\infty}\mathbb Z$$
There are many possible ring structures on this abelian group. As an example, notice that $\mathbb Z[X]$, the ring of polynomials with coefficients in $\mathbb Z$, has the same additive structure, and contains no zero divizors. Thus, we can transfer the multiplication from $\mathbb Z[X]$ to $\mathbb Q _+$.
