Is there a known example of a ring endomorphism $f: \mathbb{Z}[x_1, \ldots, x_n] \to \mathbb{Z}[x_1, \ldots, x_n]$ such that $f \circ f = f$ but whose image is not isomorphic to a polynomial ring?
My own interest in this has to do with better understanding the (categorical) Cauchy completions of Lawvere theories for some familiar types of algebraic objects; here we are dealing with the Lawvere theory of commutative rings. But apparently this *type* of problem is of interest to algebraists in the context of hard problems like the Jacobian conjecture and the cancellation problem, so there is a certain body of work out there already on related material.
From the literature I've scanned so far (articles by Costa, Shpilrain, Picavet, Gupta, and others), a lot of attention is paid to retracts of polynomial algebras over *fields*, but I'm having quite a hard time finding a clear statement for the case of polynomial algebras over $\mathbb{Z}$. One tantalizing lead was a statement by Picavet <a href="http://math.univ-bpclermont.fr/~picavet/archives/2002-30.pdf">here</a>: "We were motivated by an unsolved conjecture: a projective algebra of finite type over a field $A$ is a polynomial ring. An example by Costa shows that the statement is false if $A$ is not a field." I couldn't find a statement by Costa which treated every non-field $A$ (including $\mathbb{Z}$ in particular); I suspect Picavet meant that there *exist* non-fields $A$ for which the statement is false. An interesting example of such $A$ can be extracted from Gupta's later negative solution to the cancellation problem, as mentioned by Jeremy Rickard <a href="http://mathoverflow.net/a/208507/2926">here</a> at the MO discussion http://mathoverflow.net/questions/90734/is-a-retract-of-a-free-object-free. (Actually, Gupta's constructions more significantly show that the statement is false for any field of positive characteristic, but this work can be used to derive some non-field examples as well.)
Akhil Mathew asked essentially the same question at Math.SE <a href="https://math.stackexchange.com/questions/98210/are-projective-rings-over-mathbbz-free">here</a>, and got pointers to literature from Mariano Suárez-Alvarez, but I'm hoping to get something more definitive now.