feat (LocalRing/MaximalIdeal/Basic): add LocalRing.not_mem_maximalIdeal and nilradmax_localization_IsSelf #17549
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PR summary 3bf55718c2Import changes for modified filesNo significant changes to the import graph Import changes for all files
Declarations diff
You can run this locally as follows## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>
## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>The doc-module for |
mbkybky
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Oct 9, 2024
| Let `S` be the localization of a commutative semiring `R` at a submonoid `M` that does not | ||
| contain 0. If the nilradical of `R` is maximal then there is a ring isomorphism between `R` and `S`. | ||
| -/ | ||
| noncomputable def nilradmax_localization_IsSelf (h : (nilradical R).IsMaximal) (h' : (0 : R) ∉ M) |
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The naming convention requires defs to be in lowerCamelCase, i.e., nilradMaxLocalizationIsSelf in this case.
| An element `x` of a commutative local semiring is not contained in the maximal ideal | ||
| iff it is a unit. | ||
| -/ | ||
| theorem LocalRing.not_mem_maximalIdeal [LocalRing R] (x : R) : |
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I think maybe this theorem should be moved to immediately follow the theorem mem_maximalIdeal.
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Co-authored-by: Yongle Hu <[email protected]>
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For theorem LocalRing.not_mem_maximalIdeal: An element
xof a commutative local semiring is not contained in the maximal ideal if and only if it is a unit.For noncomputable def nilradmax_localization_IsSelf: Let
Sbe the localization of a commutative semiringRat a submonoidMthat does not contain 0. If the nilradical ofRis maximal then there is a ring isomorphism betweenRandS.As discussed on Zulip here: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/PR.20Permission/near/472582165