# Examples of commutative rings where the prime subring is not direct summand?

My question consists almost in the title. My motivation is the study of some tensor products $A\otimes_\mathbb{Z} B$. For a (commutative) ring, let us call prime subring the subring generated by $1$ i.e. $P(A)=\mathbb{Z}.1\simeq \mathbb{Z}/c\mathbb{Z}$ ($c=char(A)\in \mathbb{N}$, being the characteristic of $A$). In all examples I know, one has $$A=P(A)\oplus M$$ as abelian groups . This is trivial in case $char(A)$ is a prime number as $A$ is, in fact, a vector space. What happens in the other cases ? In case of general results, please give reference or proof.

The prime ring of $\mathbb Q$ is not a direct summand.