# Localize(M, R)ΒΆ

- M: Module R
- R: CommutativeRing

Localize(`M`

, `R`

) produces fractions with numerators from an `R`

module `M`

and denominators being the nonzero elements of `R`

.

- 0: %
- from AbelianMonoid
- *: (%, R) -> %
- from RightModule R
- *: (Integer, %) -> %
- from AbelianGroup
- *: (NonNegativeInteger, %) -> %
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup
- *: (R, %) -> %
- from LeftModule R
- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> %
- from AbelianGroup
- -: (%, %) -> %
- from AbelianGroup

- /: (%, R) -> %
`x / d`

divides the element`x`

by`d`

.

- /: (M, R) -> %
`m / d`

divides the element`m`

by`d`

.- <: (%, %) -> Boolean if M has OrderedAbelianGroup
- from PartialOrder
- <=: (%, %) -> Boolean if M has OrderedAbelianGroup
- from PartialOrder
- =: (%, %) -> Boolean
- from BasicType
- >: (%, %) -> Boolean if M has OrderedAbelianGroup
- from PartialOrder
- >=: (%, %) -> Boolean if M has OrderedAbelianGroup
- from PartialOrder
- ~=: (%, %) -> Boolean
- from BasicType
- coerce: % -> OutputForm
- from CoercibleTo OutputForm

- denom: % -> R
`denom x`

returns the denominator of`x`

.- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- latex: % -> String
- from SetCategory
- max: (%, %) -> % if M has OrderedAbelianGroup
- from OrderedSet
- min: (%, %) -> % if M has OrderedAbelianGroup
- from OrderedSet

- numer: % -> M
`numer x`

returns the numerator of`x`

.- opposite?: (%, %) -> Boolean
- from AbelianMonoid
- sample: %
- from AbelianMonoid
- smaller?: (%, %) -> Boolean if M has OrderedAbelianGroup
- from Comparable
- subtractIfCan: (%, %) -> Union(%, failed)
- from CancellationAbelianMonoid
- zero?: % -> Boolean
- from AbelianMonoid

BiModule(R, R)

Comparable if M has OrderedAbelianGroup

Module R

OrderedAbelianGroup if M has OrderedAbelianGroup

OrderedAbelianMonoid if M has OrderedAbelianGroup

OrderedAbelianSemiGroup if M has OrderedAbelianGroup

OrderedCancellationAbelianMonoid if M has OrderedAbelianGroup

OrderedSet if M has OrderedAbelianGroup

PartialOrder if M has OrderedAbelianGroup