# LeftModule RΒΆ

catdef.spad line 806 [edit on github]

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

- 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

- zero?: % -> Boolean if R has AbelianMonoid
from AbelianMonoid

AbelianGroup if R has AbelianGroup

AbelianMonoid if R has AbelianMonoid

CancellationAbelianMonoid if R has AbelianGroup