Why isn't the set $\mathbb{R}^{+}$ of all nonnegative real numbers with the usual operations of + and · a ring?

Question

The way I see it $(\mathbb{R_+})$ has an identity element and a zero element as well as it is closed on addition and multiplication. Yet my textbook lists it explicitly as a non example of a ring.

Answer 1

On a ring, the additive operation must make an abelian group. You don't have additive inverses in your structure, so it is not a ring.

In jest, the kind of structure you have here (where the addition makes a commutative monoid rather than an abelian group) is some times called a rig: It's a ring without negatives, so its name is ring without the n (a more conventional name is semiring). In the same vein, there is what some people call rng: it's a ring without (multiplicative) identity (more conventionally called a non-unital ring).

Answer 2

$\mathbb R_+$ is a monoid with respect to addition, but it doesn't have an inverse (since that would be the negative numbers). So that fails one of the ring conditions.