Let $A$ be a ring over field $k$, let us denote $A\otimes_k A$ by $A^e$. In Jacobson Basic Algebra II, Page 375, Exercise 6.11, it was asked to prove that if $A^e$ has an idempotent $x$ such that $(a\otimes1)x=(1\otimes a)x$ for all $a\in A$, then $A$ is $A^e$ projective.
For projectivity, it suffices to show that $A$ is a summand of $A^e$ as $A^e$-module.
We can always define the splitting $s$ to the augmentation map $\epsilon\colon A^e\to A, \sum a_i\otimes b_i\to \sum a_ib_i$ by $s(\epsilon(a))=ax$. The previous equalities ensure that $s$ is well defined. However it is unclear how to show $\epsilon s=1_A$.