Can a ring without unity have a non-trivial subring with unity? The original problem was to prove that for any commutative ring R, there is a canonical (unique) ring homomorphism from Z to R. The problem would be simple, if R has $1_R$ and we can let the homomorphism map $\phi$ sends $0$ to $0_R$ and $1$ to $1_R$. But if R doesn't have an identity, and $\phi(Z)$ forms a subring of $R$ with unity $\phi(1)$, then the Rng $R$ has a subring with unity. If ring without unity can only have a trivial subring with unity, $\phi$ must be the trivial map, and the proof would be complete. But I'm not sure of this part. Any help would be appreciated!

Many "commutative rings" are assumed from context to contain the identity, and homomorphisms of these rings must take the identity to the identity. If that is the case, define $\phi:\mathbb{Z}\to R$ so that $\phi(n)=1_R+...+1_R$ (n times).
If this is not assumed, it is false that for any commutative ring $R$ there exists a unique ring homomorphism $\phi:\mathbb{Z}\to R.$ For example if $R=\mathbb{Z}$ either the trivial map or the identity map is a homomorphism in the non-unital ring sense.
