I would like to understand the structure of the subrings A of the ring of formal power series k[[t]] (where k is a field) which satisfy the condition (A : k[[t]]) $\neq$ {0} and k $\subset$ A. Are they of the form {a$\in$ k[[t]] | v(a)$\geq$ n} + k ?
v is the order of the formal power series and (A : k[[t]]) = {a $\in$ k[[t]] | a k[[t]] $\subset$ A}. Would someone help me with that?