The ring $K[[x_1,ldots,x_n]][x_1^-1,ldots,x_n^-1]$ is UFD?

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It is well known that the ring of formal power series over a field $K[[x_1,x_2,ldots,x_n]]$ is an UFD. My question is the following: the ring $K[[x_1,ldots,x_n]][x_1^-1,ldots,x_n^-1]$ is also UFD? I was not able to find any reference on such a result.




commutative-algebra power-series unique-factorization-domains




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  • 2




    Yes, it is a UFD because the localization of a UFD is also a UFD.
    – Jesko Hüttenhain
    21 hours ago










  • But it is a localization?! With respect to which multiplicatively closed set? Maybe the set of monomials in $K[x_1,ldots,x_n]$?
    – Mircea
    17 hours ago










  • The set of monomials will work perfectly, or you can think of it as a sequence of $n$ localizations with respect to each of the variables.
    – Jesko Hüttenhain
    15 hours ago










  • Thank you for your answer.
    – Mircea
    9 mins ago














up vote
1
down vote

favorite












It is well known that the ring of formal power series over a field $K[[x_1,x_2,ldots,x_n]]$ is an UFD. My question is the following: the ring $K[[x_1,ldots,x_n]][x_1^-1,ldots,x_n^-1]$ is also UFD? I was not able to find any reference on such a result.




commutative-algebra power-series unique-factorization-domains




share|cite|improve this question















  • 2




    Yes, it is a UFD because the localization of a UFD is also a UFD.
    – Jesko Hüttenhain
    21 hours ago










  • But it is a localization?! With respect to which multiplicatively closed set? Maybe the set of monomials in $K[x_1,ldots,x_n]$?
    – Mircea
    17 hours ago










  • The set of monomials will work perfectly, or you can think of it as a sequence of $n$ localizations with respect to each of the variables.
    – Jesko Hüttenhain
    15 hours ago










  • Thank you for your answer.
    – Mircea
    9 mins ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











It is well known that the ring of formal power series over a field $K[[x_1,x_2,ldots,x_n]]$ is an UFD. My question is the following: the ring $K[[x_1,ldots,x_n]][x_1^-1,ldots,x_n^-1]$ is also UFD? I was not able to find any reference on such a result.




commutative-algebra power-series unique-factorization-domains




share|cite|improve this question











It is well known that the ring of formal power series over a field $K[[x_1,x_2,ldots,x_n]]$ is an UFD. My question is the following: the ring $K[[x_1,ldots,x_n]][x_1^-1,ldots,x_n^-1]$ is also UFD? I was not able to find any reference on such a result.





commutative-algebra power-series unique-factorization-domains





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked 23 hours ago









Mircea

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  • 2




    Yes, it is a UFD because the localization of a UFD is also a UFD.
    – Jesko Hüttenhain
    21 hours ago










  • But it is a localization?! With respect to which multiplicatively closed set? Maybe the set of monomials in $K[x_1,ldots,x_n]$?
    – Mircea
    17 hours ago










  • The set of monomials will work perfectly, or you can think of it as a sequence of $n$ localizations with respect to each of the variables.
    – Jesko Hüttenhain
    15 hours ago










  • Thank you for your answer.
    – Mircea
    9 mins ago












  • 2




    Yes, it is a UFD because the localization of a UFD is also a UFD.
    – Jesko Hüttenhain
    21 hours ago










  • But it is a localization?! With respect to which multiplicatively closed set? Maybe the set of monomials in $K[x_1,ldots,x_n]$?
    – Mircea
    17 hours ago










  • The set of monomials will work perfectly, or you can think of it as a sequence of $n$ localizations with respect to each of the variables.
    – Jesko Hüttenhain
    15 hours ago










  • Thank you for your answer.
    – Mircea
    9 mins ago







2




2




Yes, it is a UFD because the localization of a UFD is also a UFD.
– Jesko Hüttenhain
21 hours ago




Yes, it is a UFD because the localization of a UFD is also a UFD.
– Jesko Hüttenhain
21 hours ago












But it is a localization?! With respect to which multiplicatively closed set? Maybe the set of monomials in $K[x_1,ldots,x_n]$?
– Mircea
17 hours ago




But it is a localization?! With respect to which multiplicatively closed set? Maybe the set of monomials in $K[x_1,ldots,x_n]$?
– Mircea
17 hours ago












The set of monomials will work perfectly, or you can think of it as a sequence of $n$ localizations with respect to each of the variables.
– Jesko Hüttenhain
15 hours ago




The set of monomials will work perfectly, or you can think of it as a sequence of $n$ localizations with respect to each of the variables.
– Jesko Hüttenhain
15 hours ago












Thank you for your answer.
– Mircea
9 mins ago




Thank you for your answer.
– Mircea
9 mins ago















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