Gelanggang homomorfisme
Struktur aljabar
Struktur terkait
Teori bilangan aljabar
Teori bilangan dan desimal p-adik
Geometri aljabar
Geometri aljabar nonkomutatif
Aljabar bebas
Aljabar Clifford
Dalam matematika, modul adalah suatu struktur aljabar dasar yang digunakan dalam aljabar abstrak. Modul diatas gelanggang meru generalisasi dari gagasan ruang vektor diatas medan, dimana skalar sesuai apabila elemen dari gelanggang yang diberikan secara sembarang (dengan identitas) dan perkalian (di kiri dan/atau di kanan) didefinisikan antara elemen gelanggang dan elemen modul. Modul mengambil skalar dari gelanggang R disebut modul-R.
Jadi, modul sebagai ruang vektor, adalah aditif grup abelian; produk didefinisikan antara elemen gelanggang dan elemen modul distributif selama operasi penambahan setiap parameter dan kompatibel dengan perkalian gelanggang.
Modul sangat erat kaitannya dengan teori wakilan dari grup. Dan juga merupakan salah satu pengertian sentral aljabar komutatif dan aljabar homologis, dan digunakan secara luas dalam geometri aljabar dan topologi aljabar.
Dalam ruang vektor, himpunan skalar adalah medan dan bekerja pada vektor dengan perkalian skalar, apabila aksioma tertentu seperti hukum distributif. Dalam modul, skalar digunakan gelanggang, jadi konsep modul mewakilan generalisasi yang signifikan. Dalam aljabar komutatif, ideal dan gelanggang hasil bagi adalah modul, sehingga banyak argumen tentang ideal atau gelanggang hasil bagi yang menggabungkan satu argumen tentang modul. Dalam aljabar non-komutatif, perbedaan antara ideal kiri, ideal, dan modul menjadi lebih jelas, meskipun beberapa kondisi teori gelanggang apabila diekspresikan baik tentang ideal kiri atau modul kiri.
Sebagian besar teori modul terdiri dari perluasan sebanyak mungkin properti ruang vektor yang diinginkan ke ranah modul melalui gelanggang, seperti prinsip ideal. Namun, modul bisa sedikit lebih rumit daripada ruang vektor; misalnya, tidak semua modul memiliki basis, dan bahkan yang memiliki, modul bebas, tidak perlu memiliki peringkat unik jika gelanggang yang mendasarinya tidak memenuhi kondisi bilangan basis invarian, tidak seperti ruang vektor, yang selalu memiliki basis (mungkin tak hingga) yang kardinalitasnya kemudian unik. Dua pernyataan terakhir ini membutuhkan aksioma pilihan secara umum, tetapi tidak dalam kasus ruang berdimensi hingga, atau ruang berdimensi tak hingga tertentu yang berperilaku baik seperti Ruang Lp.
Misalkan R adalah gelanggang dan 1 adalah identitas perkaliannya. Modul kiri-R pada M yang terdiri dari grup abelian (M, +) dan operasi ⋅ : R × M → M maka r, s di R dan x, y di M, memiliki:
Pengoperasian gelanggang pada M disebut perkalian skalar, dan biasanya ditulis dengan penjajaran, yaitu sebagai rx untuk r pada R dan x pada M, meskipun dilambangkan sebagai r ⋅ x untuk membedakannya dari operasi perkalian gelanggang, yang dilambangkan dengan penjajaran. Notasi RM menunjukkan modul kiri-R pada M. Sebuah modul kanan-R pada M atau MR didefinisikan serupa, kecuali bahwa gelanggang itu bekerja di sebelah kanan; yaitu, perkalian skalar mengambil bentuk ⋅ : M × R → M, dan aksioma atas ditulis dengan skalar r dan s sebelah kanan x dan y.
Penulis yang tidak memerlukan gelanggang menjadi unital untuk menghilangkan ketentuan 4 atas dalam definisi modul R, dan apabila struktur yang didefinisikan atas "unital kiri R". Dalam artikel ini, sesuai dengan glosarium teori gelanggang, semua gelanggang dan modul dianggap tidak sama.[1]
Misalkan M adalah modul-R kiri dan N adalah subgrup dari M. Maka N adalah submodul (atau lebih eksplisit R) apabila n pada N dan r pada R, produk r ⋅ n adalah N (atau n ⋅ r untuk modul-R.
Jika X adalah himpunan bagian dari modul-R, maka submodul yang direntang oleh X didefinisikan sebagai ⟨ ⟨ --> X ⟩ ⟩ --> = ⋂ ⋂ --> N ⊇ ⊇ --> X N {\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N} , dimana N submodul atas dari M yang berisi X, atau secara eksplisit { ∑ ∑ --> i = 1 k r i x i ∣ ∣ --> r i ∈ ∈ --> R , x i ∈ ∈ --> X } {\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}} , yang terpenting dalam definisi adalah produk tensor.[2]
Himpunan submodul dari modul tertentu M, bersama dengan dua operasi biner + dan ∩, dalam bentuk sebuah kekisi yang memenuhi hukum modular: Diberikan submodul U, N1, N2 dari M sedemikian rupa sehingga N1 ⊂ N2, maka dua submodul berikut ini: (N1 + U) ∩ N2 = N1 + (U ∩ N2).
Jika M dan N misal modul R, maka sebuah peta f : M → N adalah homomorfisme dari modul-R jika untuk setiap m, n dalam M dan r, s dalam R,
Homomorfisme ini objek matematika lainnya, hanyalah pemetaan dengan mempertahankan struktur objek. Nama lain untuk homomorfisme modul R adalah peta linear-R.
Sebuah bijektif modul homomorfisme f : M → N disebut modul isomorfisme, dan dua modul M dan N disebut isomorfik. Dua modul isomorfik identik untuk semua tujuan praktis, hanya berbeda dalam notasi untuk elemennya.
Kernel dari modul homomorfisme f : M → N adalah submodul dari M yang terdiri dari semua elemen urutan ke nol oleh f, dan citra dari f adalah submodul dari N yang terdiri dari nilai f(m) untuk semua elemen m dari M.[3] Teorema isomorfisme yang familiar dari grup dan ruang vektor valid untuk modul-R.
Diberikan gelanggang-R, himpunan semua modul kiri-R bersama dengan homomorfisme modul dalam bentuk kategori abelian, dilambangkan dengan Mod-R(lihat kategori modul).
Jika R {\displaystyle R} adalah gelanggang komutatif dan A {\displaystyle A} adalah aljabar asosiatif-R, maka adalah modul kiri- A {\displaystyle A} dengan modul- R {\displaystyle R} pada M {\displaystyle M} bersama dengan modul homomorfisme- R {\displaystyle R}
dirumuskan sebagai
gilt.
Modul kanan- A {\displaystyle A} adalah modul- R {\displaystyle R} pada M {\displaystyle M} bersama dengan modul homomorfisme- R {\displaystyle R}
Modul gabungan dan bimodul didefinisikan secara analogi dengan kasus gelanggang.
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