maximal ideals of ring of formal power series
Suppose that is a commutative ring with non-zero unity.
If is a maximal ideal of , thenβ β is a maximal ideal of the ring of formal power series.
Also the converse is true, i.e. if is a maximal ideal of , then there is a maximal ideal
of such thatβ .
Note.β In the special case that is a field, the only maximal ideal of which is the zero ideal , this corresponds to the only maximal ideal of (see http://planetmath.org/node/12087formal power series over field).
We here prove the first assertion.β So, is assumed to be maximal.β Let
be any formal power series in .β Hence, the constant term cannot lie in .β According to the criterion for maximal ideal, there is an element of such thatβ .β Therefore
whence the same criterion says that is a maximal ideal of .
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 |