MATRIKS BERSIH KUAT ATAS RING DERET PANGKAT TERGENERALISASI MIRING
DOI:
https://doi.org/10.25077/jmu.10.3.385-393.2021Abstract
Salah satu konsep dalam teori aljabar yang banyak digunakan adalah matriks atas lapangan (field). Dalam perkembangannya, konsep matriks atas lapangan diperumum menjadi matriks atas ring. Ring merupakan suatu sistem matematika yang terdiri dari suatu himpunan tak kosong yang dilengkapi dua operasi biner yang memenuhi beberapa aksioma. Ring yang banyak digunakan dalam kajian ilmu matematika terapan adalah Ring Polinomial R[X] dan Ring Deret Pangkat R[[X]]. Salah satu sifat matriks atas ring yang telah dikaji oleh para peneliti adalah syarat cukup matriks atas ring R[[X]] merupakan matriks bersih kuat. Pada perkembangannya, struktur R[[X]] digeneralisasi menjadi ring semigrup R[S], Ring Deret Pangkat Tergeneralisasi (RDPT) [[RS,≤]], dan Ring Deret Pangkat Tergeneralisasi Miring (RDPTM) R[[S, ≤, ω]]. Berdasarkan fakta bahwa struktur R[[S, ≤, ω]] lebih umum dari R[[X]], pada penelitian ini diberikan syarat cukup matriks atas RDPTM R[[S, ≤, ω]] merupakan matriks bersih kuat. Hal ini dapat dilakukan dengan cara menambahkan beberapa syarat pada struktur ring R, monoid terurut tegas (S, ≤), dan homomorfisma monoid ω sehingga matriks atas R[[S, ≤, ω]] merupakan matriks bersih kuat. Sebagai akibat langsung, hasil penelitian ini lebih umum dari syarat cukup matriks atas R[[X]] merupakan matriks bersih kuat yang telah dikaji sebelumnya.
Kata Kunci: Matriks atas ring, matriks bersih kuat, ring deret pangkat tergeneralisasi miring
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