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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.

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

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  • $\begingroup$ Excellent ! I should have thought of this myself. Thank you. I vote +1. Remains the quest of results in this direction: I am specially interested by the positive non-prime characteristic. $\endgroup$ – Duchamp Gérard H. E. Mar 25 '15 at 16:26

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