Commutative semiring such that every element except zero does not have additive inverse and each element can be uniquely sum-decomposed

Is there a unique factorization commutative semiring such that every element except zero does not have additive inverse in the semiring and each element can be decomposed into unique finite sum decomposition?

That is, for every $x$ in the element, it has unique sum decomposition into $k \geq 1$ elements, $x = a_1 + ... + a_k$ and each $a_i$ can be uniquely factorizable into prime elements. There does not necessarily need to be multiplicative identity.

• Doesn't the zero always have an inverse? – Jakob Werner Aug 29 '14 at 6:26
• @JakobWerner Oh. i was meaning non-zero element. Corrected. – nola Aug 29 '14 at 6:28

In the semiring $(\mathbb{N}, +, \times)$, the additive component is the free monoid on one generator. Thus $1$ is the unique irreducible element, and each element has a unique decomposition as a sum of $1$, like $3 = 1 + 1 + 1$. No element except zero has an inverse.