Can there be an onto homomorphism from a ring without unity to a ring with unity?

I'm surprised nobody suggested this obvious construction yet:

Let $R_1$ be a ring without identity, and $R_2$ be a ring with identity. Then $R=R_1\times R_2$ is a ring without identity, and $(r_1,r_2)\mapsto r_2$ is a surjective ring homomorphism of $R\to R_2$.

Let $R$ be the set of even integers. This is a ring without unity.

Let $R'$ be the ring of integers modulo $3$. This is a ring with unity.

Let $\phi:R\to R'$ be the "reduction modulo $3$" map. This is a surjective homomorphism of rings.

Algebras of functions are a source of examples. Consider the restriction homomorphism from $C_0(\mathbb R)$ onto $C[0,1]$

Remark All of the examples given up to now, maybe with exception of that of Jendrik Stelzner, have the same feel to me. E.g., the example of Lord Shark: associate to an integer $n$ a "function" on the primes whose value at $p$ is $[n]_p$. Lord Shark takes the subring of the integers whose value at $p = 2$ is zero, and evaluates at $p = 3$. It's easy (but probably pointless) to fit rschweib's example into this rubric. My other example with irreducible representations of a compact group also fits the pattern: associate to an element $f$ of the convolution algebra a "function" $\hat f$ on the set of irreducible representations, defined by $\hat f(\pi) = \pi(f)$. This is a sort of Fourier transform. The set of functions you get will satisfy a $c_0$ condition, and in any case has no identity element. Now the homomorphism evaluates these functions at a finite point to produce a ring with identity.