# Documentation

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# `Cat`::`LeftModule`

Category of left R -modules

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## Syntax

```Cat::LeftModule(`R`)
```

## Description

`Cat::LeftModule(R)` represents the category of left `R`-modules.

A `Cat::LeftModule(R)` is an Abelian group together with a rng `R` (a ring without unit) and a left multiplication `*` (`_mult`).

The left multiplication is an operation taking an element of rng `R` and a module element and returning a module element.

Given ring elements a, b and module elements x, y the following 3 distibutive laws must hold:

1. (ab) x = a (bx),

2. (a + b) x = ax + bx,

3. a (x + y) = ax + ay.

Beware: The operation of a non-Abelian semi-group is also written as `*` (`_mult`). The method `"_mult"` must handle the situation if a left module is also a non-Abelian semi-group. In such a case it must both implement the group operation and the left multiplication by elements of the rng.

## Categories

`Cat::AbelianGroup`

## Parameters

 `R` A domain which must be from the category `Cat::Rng`.

## Methods

expand all

#### Basic Methods

`_mult(r, x)`