Let K be a field. Show that K[[x]] (the formal power series ring with coefficients in K) has a unique maximal ideal.

Attempt at a solution: Let I $\subset$ K[[x]] with I $\neq$ K[[x]] and I= ($x$). So for any element $a$ $\in$K[[x]] such that $a$ $\not\in$ I, that is $a$ is a polynomial with degree zero (constant), then I + A, where A= ($a$), is equal to K[[x]]. This is because any polynomial with constant term equal to zero, is generated by $x$. Hence in I. Thus the ideal of constant terms plus I is equal to K[[x]] (A+I=K[[x]]). This is sufficient to say that I is maximal.

Since the quotient ring $\dfrac{K[[x]]}{I}$ has only trivial ideals (K field), then there is no other maximal ideal besides I. Hence it is unique.

Any help would be appreciated.


Hint: a ring has a unique maximal ideal if and only if the set of non invertible elements is an ideal; what are the non invertible elements in $K[[x]]$?


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