When a ring is a polynomial ring?

In the paper (2.11) the authors show that if k is a separable algebraic extension of k and x1,x2,,xn are indeterminates over k and a normal one dimensional ring A with kAk[x1,x2,,xn] then A has the form k[t] where k is the algebraic closer of k in A.

The above is a very strict sufficient condition to ensure the polynomial property. Is there a general answer to the question:

Q. When a ring R is a polynomial ring over a field?


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