maximal ideals of ring of formal power series

Suppose that R is a commutative ring with non-zero unity.

If π”ͺ is a maximal idealMathworldPlanetmath of R, then  𝔐:=π”ͺ+(X)  is a maximal ideal of the ring R⁒[[X]] of formal power series.

Also the converse is true, i.e. if 𝔐 is a maximal ideal of R⁒[[X]], then there is a maximal ideal π”ͺ of R such that  𝔐=π”ͺ+(X).

Note.  In the special case that R is a field, the only maximal ideal of which is the zero idealMathworldPlanetmathPlanetmath (0), this corresponds to the only maximal ideal (X) of R⁒[[X]] (see power series over field).

We here prove the first assertion.  So, π”ͺ is assumed to be maximal.  Let


be any formal power series in R⁒[[X]]βˆ–π”.  Hence, the constant term a0 cannot lie in π”ͺ.  According to the criterion for maximal ideal, there is an element r of R such that  1+r⁒a0∈π”ͺ.  Therefore


whence the same criterion says that 𝔐 is a maximal ideal of R⁒[[X]].

Title maximal ideals of ring of formal power series
Canonical name MaximalIdealsOfRingOfFormalPowerSeries
Date of creation 2013-03-22 19:10:49
Last modified on 2013-03-22 19:10:49
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Result
Classification msc 13H05
Classification msc 13J05
Classification msc 13C13
Classification msc 13F25