# LeftModule RΒΆ

- R: SemiRng

The category of left modules over an `rng`

(ring not necessarily with unit). This is an abelian group which supports left multiplication by elements of the `rng`

.

- 0: % if R has AbelianMonoid
- from AbelianMonoid
- *: (Integer, %) -> % if R has AbelianGroup
- from AbelianGroup
- *: (NonNegativeInteger, %) -> % if R has AbelianMonoid
- from AbelianMonoid
- *: (PositiveInteger, %) -> %
- from AbelianSemiGroup

- *: (R, %) -> %
`r*x`

returns the left multiplication of the module element`x`

by the ring element`r`

.- +: (%, %) -> %
- from AbelianSemiGroup
- -: % -> % if R has AbelianGroup
- from AbelianGroup
- -: (%, %) -> % if R has AbelianGroup
- from AbelianGroup
- =: (%, %) -> Boolean
- from BasicType
- ~=: (%, %) -> Boolean
- from BasicType
- coerce: % -> OutputForm
- from CoercibleTo OutputForm
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- latex: % -> String
- from SetCategory
- opposite?: (%, %) -> Boolean if R has AbelianMonoid
- from AbelianMonoid
- sample: % if R has AbelianMonoid
- from AbelianMonoid
- subtractIfCan: (%, %) -> Union(%, failed) if R has AbelianGroup
- from CancellationAbelianMonoid
- zero?: % -> Boolean if R has AbelianMonoid
- from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid if R has AbelianMonoid

CancellationAbelianMonoid if R has AbelianGroup