THE SUFFICIENT AND NECESSARY CONDITIONS FOR A MODULE TO BE A WEAKLY UNIQUE FACTORIZATION MODULE

Prabhadika, I Putu Yudi
Wahyuni, Sri

BAREKENG: Jurnal Ilmu Matematika dan Terapan
Vol 19 No 1 (2025): BAREKENG: Journal of Mathematics and Its Application

T A torsion-free module over an integral domain is called Unique Factorization Module (UFM) if satisfied some conditions: (1) Every non-zero element has an irreducible factorization, that is , with are irreducible in and is irreducible in , and (2) if are two irreducible factorizations of , then in , and we can rearrange the order of the ’s so that in for every . The definition of UFM is a generalization of the concept of factorization on the ring which is applied to the module. In this study, we will discuss another definition that is a generalization of UFM, namely by the Weakly Unique Factorization Module (w-UFM). First, some concepts that play an important role in defining w-UFM are given. After that, the definition and characterization of w-UFM is also given. The results of this study will provide the sufficient and necessary conditions of the w-UFM.