units in discrete valuation rings

Let $B/A$ be an extension of discrete valuation rings such that the purely inseparable extension of residue fields $l/k$ has a primitive element, i.e., $l=k(y)$ for some element $y$ in $l$. I want to know if one can plug in another discrete valuation ring as follows.

Let $x$ be a lift of $y$ to $B$. Is $A[x]$ a dvr? What are necessary and sufficient conditions?

Of course, the answer is yes if $e=1$. Then $A[x] = B$.

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I presume you mean "let $x$ be a lift of $y$ to $B$"? –  David Loeffler Jan 28 '12 at 9:42