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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?

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No, they are not. Consider the $k$-algebra $k \oplus t^2.k \oplus t^4.k[[t]]$. –  user10676 May 2 '12 at 19:53
    
what do you mean? the power series with only even powers? it doesn't satisfy (A : k[[t]]) $\neq$ {0} –  user30486 May 2 '12 at 20:06
    
No, I mean powers series with powers greater than $4$ or equal to $2$ or $0$. This contains every power series with valuation greater than $4$. –  user10676 May 2 '12 at 23:23
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