OpenLLM-France/wikimedia huggingface dataset:
# Pinyin/HFE hereditary hemochromatosis

HFE hereditary hemochromatosis is a genetic disease. Sufferers over absorb iron, causing the total iron concentration inside the body to reach pathological level. Humans and animals have no way to excrete excess iron. Excess iron will accumulate inside tissues and organs, affecting their normal functions. The easiest affected organs are: liver, adrenal gland, heart and pancreas. Sufferers might suffer from cirrhosis, adrenal gland malfunction, heart failure, diabetes, or arthritis. This kind of illness is most common in northern European ancestry, particularly in Irish people.
HFE yíchuánxìng xuèsèbìng shì yīzhǒng yíchuánbìng. Huànzhě guòdù shèqǔ tiězhì, lìng shēntǐ nèi de zǒng tiězhì hánliàng dázhì bìnglǐxìng de shuǐpíng. Rénlèi hé qítā dòngwù dōu méiyǒu fāngfǎ páichū duōyú de tiězhì. Duōyú de tiězhì huì jījù zài zǔzhī hé qìguān nèi, yǐngxiǎng tāmen de zhèngcháng yùnzuò. Zuì róngyì shòu yǐngxiǎng de qìguān bāokuò gānzàng, shènshàngxiàn, xīnzàng hé yízàng. Huànzhě kěnéng chūxiàn gānyìnghuà, shènshàngxiàn gōngnéng-bùquán, xīn-shuāijié, tángniàobìng, huò guānjiéyán. Zhèzhǒng jíbìng zài Běi'ōu pǔxì zuìwéi pǔbiàn, yóuqíshì Ireland-rén.
# Pinyin/Habitat

A habitat is an area that is lived in by a particular species of animal, plant, or other type of organism. It is the natural environment in which an organism lives, or the physical environment that surrounds a population.
Qīxīdì shì dòngwù, zhíwù huò qítā lèixíng shēngwù de tèdìng wùzhǒng de shēnghuó qūyù. Tā shì yīgè shēngwùtǐ huò yīgè zúqún shēnghuó de zìrán huánjìng.
# Pinyin/Haikou

Search Haikou
Haikou is the capital of Hainan province. It is located at the northeast of Hainan, the mouth of the Nandu river, and the entrance of railway ferries. The north of Haikou is separated by Qiongzhou strait, and facing Leizhou peninsula of Guangdong province. It has an area of 2,304.84km$^2$, with a population of 2,100,000+.
Haaikoou shih Haainarn Sheeng sheenghuih. ta weihyur Haainarn dongbeeibuh, Narnduh Jiang ruhhaaikoou, tieeluh duhlurn ruhkoou. Haaikoou beeimiahn xiangger Qiorngzhou Haaixiar yuu Guaangdong Sheeng d Leirzhou Bahndaao xiangwahng. ta d miahnji 2304.84 km$^2$, rernkoou 210'0000+.
# Pinyin/Television
## Noun

* a television is a piece of electrical equipment with a screen, used for watching programs.
** diahnshih shih yoou pirngmuh d diahnqih, yohnglair kahn jiermuh.
# Pinyin/Ten Thousand

See also
# Pinyin/Tendon

A tendon is a tough band of fibrous connective tissue that usually connects muscle to bone and is capable of withstanding tension. Tendons are similar to ligaments and fasciae; all three are made of collagen. Ligaments join one bone to another bone; fasciae connect muscles to other muscles. Tendons and muscles work together to move bones.
jinjiahn shì jiānrèn de dàizhuàng xiānwéi liánjiēxìng zǔzhī, tōngcháng liánjiē jīròu zhì gǔtou, bìng néng chéngshòu zhānglì. Jīnjiàn lèisì rèndài hé jīnmó, dōu shì yóu jiāoyuándànbái zǔchéng. Rèndài liánjiē gǔtou zhì gǔtou, jīnmó liánjiē jīròu zhì jīròu. Jīnjiàn hé jīròu gòngtóng yídòng gǔtou.
# Pinyin/Technology
## Noun

* technology: advanced scientific knowledge used for practical purposes
** kejih: yohngyur shirjih muhdih d xianjihn kexuer zhishir
# Pinyin/Telecom

See also
# Pinyin/Telecommunication

Telecommunication is the way of communication using electromagnetic waves.
Diànxìn shì lìyòng diàncíbō jìnxíng tōngxìn de fāngshì.
# Pinyin/Telescope

Search
# Pinyin/Temperature

Temperature is the measurement in degrees of how hot or cold a thing or place is.
wenduh shih liarngduor wuhtii huoh dihfang d leengreh d duhshuh.
# Pinyin/Temple

See also
# Pinyin/Measles

Measles is a highly contagious viral disease, which affects mostly children. It is transmitted via droplets from the nose, mouth or throat of infected persons. Initial symptoms, which usually appear 10–12 days after infection, include high fever, runny nose, bloodshot eyes, and tiny white spots on the inside of the mouth. Several days later, a rash develops, starting on the face and upper neck and gradually spreading downwards.
There is no specific treatment for measles and most people recover within 2–3 weeks. However, particularly in malnourished children and people with reduced immunity, measles can cause serious complications, including blindness, encephalitis, severe diarrhoea, ear infection and pneumonia. Measles can be prevented by immunization.
Mázhěn shì 1-zhǒng jùyǒu gāodù chuánrǎnxìng de bìngdúxìng jíbìng, shòu yǐngxiǎng de dàbùfèn shì értóng. Mázhěn bìngdú tōngguò shòugǎnrǎnzhě kǒu, bí, hóu de fēimò chuánbō. Chūqī zhèngzhuàng tōngcháng zài bèi gǎnrǎn 8-12 tiān zhīhòu xiǎnxiàn, bāokuò gāoshāo, liútì, yǎnjīng chōngxuè hé kǒu nèi chūxiàn báisè xiǎo diǎn. Ruògān tiānhòu, liǎnbù hé shàng jǐngbù kāishǐ chūxiàn pízhěn, bìng zhúbù xiàngxià mànyán.
Mázhěn méiyǒu tèyìxìng zhìliáo fāngfǎ, dàduōshù rén zài fābìng hòu de 2-3 zhōu nèi kāngfù. Rán'ér, yóuqíshì zài yíngyǎng bùliáng de értóng hé miǎnyìlì dīxià de rénqún zhōng, mázhěn kě yǐnqǐ yánzhòng de bìngfāzhèng, bāokuò shīmíng, nǎoyán, yánzhòng fùxiè, ěrbù gǎnrǎn hé fèiyán. Mázhěn kě tōngguò miǎnyì jiēzhòng dédào yùfáng.
# Pinyin/Measurement

A measurement is the act or the process of finding the size, quantity or degree of something.
duhliahng shih zhaaochu moouwuh chiicuhn, shuhliahng huoh duhshuh d xirngweir huoh gouhcherng.
# Pinyin/Meat

Meat is the flesh of animals that is used as food.
Ròulèi shì yòngzuò shíwù de dòngwù de ròu.
# Pinyin/Mean

See also
# Pinyin/Meaning

Meaning is the thing, action, feeling, idea, etc. that a word or words represent.
yihsi shih ciryuu daihbiaao d shihwuh, dohngzuoh, gaanjuer, gaihniahn deeng.
# Pokémon/Pokédex/Luxio
## Description

A quadrupedal pokémon.

## Base Statistics

A Electric type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Luxio: ground Moves that deal 50% of their damage to Luxio: electric, flying, steel
# Pokémon/Pokédex/Luvdisc
## Basic Statistics

* National Index: 370
* Johto Index: 370
* Hoenn Index: 183
* Stage: Basic
* Height: 0.6m/2'0"
* Weight: 8.7kg/19.0lb
* Gender Distribution: 75% Female
* Ability: Swift Swim
* Type: Water
* Species: Rendezvous
* Egg Group: Water 2
* Hatch Steps: 5120
* Catch Rate: 225
* Base Experience: 110
* Max Experience: 800000
| | HP | Attack | Defense | Speed | Special Attack | Special Defense |
| Base Stats (RB/Y) | - | - | - | - | - | - |
| Base Stats (GS/C) | - | - | - | - | - | - |
| Base Stats (RS/FL/E) | 43 | 30 | 55 | 97 | 40 | 65 |
| Base Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Base Stats (BW/B2W2) | – | – | – | – | – | – |
| Base Stats (XY/ORAS) | – | – | – | – | – | – |
| Base Stats (SM/USUM) | – | – | – | – | – | – |
| Base Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Max Stats (RB/Y) | - | - | - | - | - | - |
| Max Stats (GS/C) | - | - | - | - | - | - |
| Max Stats (RS/FL/E) | 290 | 174 | 229 | 322 | 196 | 251 |
| Max Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Max Stats (B/W/B2W2) | – | – | – | – | – | – |
| Max Stats (X/Y/ORAS) | – | – | – | – | – | – |
| Max Stats (SM/USUM) | – | – | – | – | – | – |
| Max Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Effort Points (Sword/Shield/BDSP) | – | – | – | – | – | – |

## Type Matching

Move types which are super effective on Luvdisc: Electric, Grass
Move types which are not very effective on Luvdisc:Fire, Ice, Steel , Water
All other move types are normal effectiveness on Luvdisc.

## Moveset

| Level | Ruby / Sapphire | Fire Red / Leaf Green |
| Start | Tackle | Tackle |
| 4 | Charm | Charm |
| 12 | Water Gun | Water Gun |
| 16 | Agility | Agility |
| 24 | Take Down | Take Down |
| 28 | Attract | Attract |
| 36 | Sweet Kiss | Sweet Kiss |
| 40 | Flail | Flail |
| 48 | Safeguard | Safeguard |

### TM/HM

* RS/FL: Attract, Blizzard, Dive, Double Team, Facade, Frustration, Hail, Hidden Power, Ice Beam, Protect, Rain Dance, Rest, Return, Safeguard, Secret Power, Surf, Toxic, Water Pulse, Waterfall

### Breeding

* RS/FL: Mud Sport, Splash, Supersonic, Water Sport

## Trading Cards

* EX Hidden Legends #40 - Luvdisc
# Pokémon/Pokédex/Lugia
## Basic Statistics

* National Index: 249
* Johto Index: 247
* Hoenn Index: 384
* Stage: Basic
* Height: 5.2m/17'1"
* Weight: 216kg/476.0lb
* Gender Distribution: Genderless
* Ability: Pressure
* Type: Psychic/Flying
* Species: Diving
* Egg Group: Unbreedable
* Hatch Steps: 30720
* Catch Rate: 3
* Base Experience: 220
* Max Experience: 1250000
| | HP | Attack | Defense | Speed | Special Attack | Special Defense |
| Base Stats (RB/Y) | - | - | - | - | - | - |
| Base Stats (GS/C) | 106 | 90 | 130 | 110 | 90 | 154 |
| Base Stats (RS/FL/E) | 106 | 90 | 130 | 110 | 90 | 154 |
| Base Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Base Stats (BW/B2W2) | – | – | – | – | – | – |
| Base Stats (XY/ORAS) | – | – | – | – | – | – |
| Base Stats (SM/USUM) | – | – | – | – | – | – |
| Base Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Max Stats (RB/Y) | - | - | - | - | - | - |
| Max Stats (GS/C) | 415 | 278 | 358 | 318 | 278 | 406 |
| Max Stats (RS/FL/E) | 416 | 306 | 394 | 350 | 306 | 447 |
| Max Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Max Stats (B/W/B2W2) | – | – | – | – | – | – |
| Max Stats (X/Y/ORAS) | – | – | – | – | – | – |
| Max Stats (SM/USUM) | – | – | – | – | – | – |
| Max Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Effort Points (Sword/Shield/BDSP) | – | – | – | – | – | – |

## Type Matching

Move types which are good on Lugia: Dark, Electric, Ghost, Ice, Rock Move types which are bad on Lugia: Grass, Psychic Move types which are very bad on Lugia: Fighting Move types which don't effect Lugia: Ground
All other move types are normal effectiveness on Lugia.

## Moveset

| Level | Gold / Silver | Crystal | Ruby / Sapphire Emerald XD / Colosseum | Fire Red / Leaf Green |
| Start | Aeroblast | Aeroblast | Whirlwind | Whirlwind |
| 11 | Safeguard | Safeguard | Safeguard | Safeguard |
| 22 | Gust | Gust | Gust | Gust |
| 33 | Recover | Recover | Recover | Recover |
| 44 | Hydro Pump | Hydro Pump | Hydro Pump | Hydro Pump |
| 55 | Rain Dance | Rain Dance | Rain Dance | Rain Dance |
| 66 | Swift | Swift | Swift | Swift |
| 77 | Whirlwind | Whirlwind | Aeroblast | Aeroblast |
| 88 | Ancientpower | Ancientpower | Ancientpower | Ancientpower |
| 99 | Future Sight | Future Sight | Future Sight | Future Sight |

### TM/HM

* GS/C/RS/FL: Blizzard, Double Team, Earthquake, Fly, Frustration, Giga Drain, Hidden Power, Hyper Beam, Iron Tail, Protect, Psychic, Rain Dance, Rest, Return, Roar, Rock Smash, Sandstorm, Shadow Ball, Steel Wing, Strength, Sunny Day, Surf, Thunder, Toxic, Waterfall
* C/RS/FL: Ice Beam, Thunderbolt
* GS/C: Curse, Detect, Dragonbreath, Dream Eater, Endure, Headbutt, Icy Wind, Mud-Slap, Nightmare, Psych Up, Sleep Talk, Snore, Swagger, Swift, Whirlpool, Zap Cannon
* RS/FL: Aerial Ace, Calm Mind, Dive, Facade, Hail, Light Screen, Reflect, Safeguard, Secret Power, Shock Wave, Skill Swap, Water Pulse

## Trading Cards

* Neo Genesis #9 - Lugia
* Neo Revelation #20 - Lugia
* Aquapolis #149 - Lugia
# Pokémon/Pokédex/Ludicolo
## Basic Statistics

* National Index: 272
* Johto Index: 272
* Hoenn Index: 21
* Stage: Stage 2
* Height: 1.5m/4'11"
* Weight: 55kg/121.0lb
* Gender Distribution: 50% Male
* Ability: Swift Swim/Rain Dish
* Type: Water/Grass
* Species: Crowned
* Egg Group: Water 1/Plant
* Hatch Steps: 3840
* Catch Rate: 45
* Base Experience: 181
* Max Experience: 1059860
| | HP | Attack | Defense | Speed | Special Attack | Special Defense |
| Base Stats (RB/Y) | - | - | - | - | - | - |
| Base Stats (GS/C) | - | - | - | - | - | - |
| Base Stats (RS/FL/E) | 80 | 70 | 70 | 70 | 90 | 100 |
| Base Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Base Stats (BW/B2W2) | – | – | – | – | – | – |
| Base Stats (XY/ORAS) | – | – | – | – | – | – |
| Base Stats (SM/USUM) | – | – | – | – | – | – |
| Base Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Max Stats (RB/Y) | - | - | - | - | - | - |
| Max Stats (GS/C) | - | - | - | - | - | - |
| Max Stats (RS/FL/E) | 364 | 262 | 262 | 262 | 306 | 328 |
| Max Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Max Stats (B/W/B2W2) | – | – | – | – | – | – |
| Max Stats (X/Y/ORAS) | – | – | – | – | – | – |
| Max Stats (SM/USUM) | – | – | – | – | – | – |
| Max Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Effort Points (Sword/Shield/BDSP) | – | – | – | – | – | – |

## Type Matching

Move types which are good on Ludicolo: Bug, Flying, Poison Move types which are bad on Ludicolo: Ground, Steel Move types which are very bad on Ludicolo: Water
All other move types are normal effectiveness on Ludicolo.

## How to Get

The only way to acquire Ludicolo is evolve Lombre with a Water Stone.

## Moveset

| Level | Ruby / Sapphire | Fire Red / Leaf Green | Emerald |
| Start | Astonish Growl Absorb Nature Power | Astonish Growl Absorb Nature Power | |

### TM/HM

* RS/FL: Attract, Blizzard, Brick Break, Bullet Seed, Dive, Double Team, Facade, Flash, Focus Punch, Frustration, Giga Drain, Hail, Hidden Power, Hyper Beam, Ice Beam, Protect, Rain Dance, Rest, Return, Rock Smash, Secret Power, Solarbeam, Strength, Sunny Day, Surf, Thief, Toxic, Water Pulse, Waterfall,

### Evolution

* RS/FL: Flail, Leech Seed, Razor Leaf, Sweet Scent, Synthesis, Water Gun

## Trading Cards

* EX Sandstorm #7 - Ludicolo
* EX Deoxys #10 - Ludicolo
* EX Deoxys #19 - Ludicolo
| Basic: Lotad | Stage 1: Lombre | Stage 2: Ludicolo |
# Pokémon/Pokédex/Luxray
## Description

A quadrupedal pokémon.

## Base Statistics

A Electric type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Luxray: ground Moves that deal 50% of their damage to Luxray: electric, flying, steel
# Pokémon/Pokédex/Lucario
## Description

Lucario is a bipedal Pokémon, with a blue and black motif.

## Base Statistics

Lucario is a dual type Pokémon, possessing both fighting and steel types.

## Type Maching

Moves that deal 200% of their damage to Lucario: fire, fighting, ground Moves that deal 50% of their damage to Lucario: dark, normal, grass, ice, dragon, steel Moves that deal 25% of their damage to Lucario: bug, rock Moves that deal 0% of their damage to Lucario: poison

## How to Evolve

Riolu evolves into Lucario.
# Pokémon/Pokédex/Lurantis
## Description

A mantis-like Pokémon.

## Base Statistics

A Grass type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Lurantis: fire, ice, poison, flying, bug Moves that deal 50% of their damage to Lurantis: water, grass, electric, ground
# Pokémon/Pokédex/Lycanroc
## Description

A doglike pokémon.

## Base Statistics

A Rock type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Lycanroc: water, grass, fighting, ground, steel Moves that deal 50% of their damage to Lycanroc: normal, fire, poison, flying
# Pokémon/Pokédex/Lumineon
## Description

A fishlike Pokémon.

## Base Statistics

A Water type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Lumineon: grass, electric Moves that deal 50% of their damage to Lumineon: fire, water, ice, steel
# Plezuro/lists

One of important types of Plezuro is the list. It is an equivalent of .NET List<object> and Java ArrayList<Object>. To create a list, just use square brackets ('[', ']'). In intermediate code it is translated to '::array'. Elements of a list can be of any type (including list), they are passed by reference, so it is possible that a list contains itself.
As you just have seen above, you can print elements of a list in many ways. The simplest way is to use the dumpl function (it is appropriate to any type in Plezuro). Another one is to use the each method. 'args' means all the arguments passed to a function (in this case a tuple (value, index)) and 'this' means the argument #0 (in this case only value).
Now, I guess, you wonder to get access to a specified element of a list (get and set its value). It is pretty simple, like in other languages use square brackets.
Lists like any other variables are passed by reference.
Now, how to sort a list?
Now, a little more about the indexing. You can access to a piece of a list in a similar way like in Ruby (using '..' operator). For example a[1..4]. The difference is in this case in Ruby it returns elements from index 1 up to index 4 but in Plezuro it returns elements from index 1 up to index 3. Another interesting feature is when using commas, you can access to disjoint pieces of a list (for example first and fourth element). Indexing may return an empty object, a single variable or a tuple. Of course, you can modify an all piece of a list.
# Podcasting

Podcasting is the so-called syndicated publishing or distribution of audio or video programmes on to the Internet. Podcasts are easy to create as all you require is a name, microphone/headphones, a topic, and a laptop or a phone you could utilize to edit your podcast then record. Podcasts are in MP3, MPEG or a similar file format. Podcast producers make their programmes available with RSS feeds, a tool that allows listeners to select programmes they wish to download or to set the software to automatically download innovative programmes as they become available.

## Podcasting for listeners

Listening to podcasts is easy and enjoyable. You can listen to any topic, at any time of the morning, day, or night, 24/7. But to listen to a podcast, you need to be able to find them. Locating podcasts by relevant topics is done through Podcast Directories.

## Contents

* Podcasting/Equipment
* Podcasting/Distribution

## External links

* The Guide to Free Podcasting How to podcast using free tools explained.
* Guide to Podcasting for windows Users Using Free/Libre and open source software.
* Apple's "tech specs" for podcasting with iTunes
* Directory of Podcasts
* Apple's Fall 2005 Podcast Creation Guide
* Podcasting at Wikiversity
* PoducateMe Podcasting Guide
# Plezuro

Plezuro is a programming language created by Piotr Sroczkowski. It enables fast development and is an open source (on license GNU/GPL).

## Contents

* Introduction
* Getting started
* Built-in types
* Let's code a little bit
* Lists
# Plezuro/built-in types

Plezuro provides some built-in classes (types). Moreover you can create your own classes. Each variable is an object of a class, like in Ruby. Therefore all variables are passes into reference. In the next chapter we will discuss the difference between reference and copying. The list of basic built-in classes with their using:
# Plezuro/introduction

The main purpose of creating of Plezuro was to enhance the software development process. It was created in 2014 while there existed many other scripting languages. However none of them had enough beautiful syntax to write the large pieces of codes without getting bored. Even Plezuro needs some improvements to achieve its main goals.
# Plezuro/let's code a little bit

Now you should try to code something in Plezuro.
# Playing a Video Game/Replaying the Game

When replaying the game there are some approaches one can follow to increase replay value.
* Increasing the difficulty.
** Through In game difficulty settings.
** Through self imposed challenges and restrictions.
* Achievement hunting
* Story analysis
* Speedrunning
# Linear Algebra over a Ring/Multilinear algebra

Definition (multilinear function):
Let $R$ be a ring, and let $M_{1},\ldots ,M_{n},K$ be $R$-modules. Then the set of $R$-multilinear functions from $M_{1}\times \cdots \times M_{n}$ to $K$ is the set
    $L(M_{1},\ldots ,M_{n},K):=\left\{f:M_{1}\times M_{n}\to K|\forall j\in [n]:\forall m_{1}\in M_{1},\ldots ,m_{n}\in M_{n},l_{j}\in M_{j},r\in R:f(m_{1},\ldots ,m_{j}+rl_{j},\ldots ,\right\}$.
Proposition (equivalent definition of tensor product of free modules using multilinear functions):
Let $R$ be a ring, and let $M_{1},\ldots ,M_{n}$ be free, finitely generated $R$-modules. Then if we alternatively define
    $M_{1}^{*}\otimes \cdots \otimes M_{n}^{*}:=L(M_{1},\ldots ,M_{n},R)$,
and let the elementary tensors be $m_{1}^{*}\otimes \cdots \otimes m_{n}^{*}(l_{1},\ldots ,l_{n}):=m_{1}^{*}(l_{1})\cdots m_{n}^{*}(l_{n})$, then the $M_{1}\otimes M_{n}$ from this definition satisfies the same universal property as the usual tensor product $M_{1}\otimes \cdots \otimes M_{n}$. In particular, the two tensor products are canonically isomorphic.
Proof: For $j\in [n]$, let $(e_{\lambda }^{j})_{\lambda \in \Lambda _{j}}$ be a basis of $M_{j}$, where $\Lambda _{j}$ is the respective finite index set. Given any $R$-module $K$ and any multilinear map $f:M_{1}\times \cdots \times M_{n}\to K$, we want a unique linear function $g:M_{1}\otimes \cdots \otimes M_{n}\to K$ such that $g\circ h=f$, where $h:M_{1}\times \cdots \times M_{n}\to M_{1}^{*}\otimes \cdots \otimes M_{n}^{*}$ is the map that sends a tuple to the respective elementary tensor. $\Box$
# Linear Algebra over a Ring/Free modules and matrices

Definition (free module):
Let $R$ be a ring, and let $S$ be an arbitrary set. Then the free $R$-module over $S$, denoted $R\langle S\rangle$, is defined to be the $R$-module whose elements are functions
    $f:S\to R$
which are zero everywhere on $S$ except on finitely many elements, together with pointwise addition and scalar multiplication.
Proposition (basis of a free module):
Let $R$ be a ring, and let $S$ be a set. Then a basis for the free $R$-module over $s$ is given by the functions
    $f_{s}:S\to R,f(t):={\begin{cases}1&t=s\\0&{\text{otherwise}}.\end{cases}}$
By abuse of notation, we will write $s$ instead of $f_{s}$. Hence, the above proposition implies that we may denote an element $m\in R\langle S\rangle$ as a sum
    $m=\sum _{s\in S}a_{s}s$,
where only finitely many $a_{s}$ are nonzero.
Proof: Let $f:S\to R$ be any function that is everywhere zero except on finitely many entries, and let $s_{1},\ldots ,s_{n}=\{s\in S|f(s)\neq 0\}$. Then we have
    $f=\sum _{k=1}^{n}f(s_{k})s_{k}$. $\Box$