中文摘要：

近年来，Rota-Baxter代数在数学和物理学中有着广泛的应用，受到越来越多的关注．自由Rota-Baxter代数分别用括号字，根树以及Motzkin路径得到了构造因子分解在代数学中是一个很重要的问题．本文主要考虑用括号字构造的自由Rota-Baxter代数，得到了自由Rota-Baxter代数中基元素的因子分解．

英文摘要：

The study of Rota-Baxter algebras has attracted quite much attentions in recent years for their broad applications in mathematics and physics. Free Rota-Baxter algebras have been constructed by bracketed words, rooted trees and Motzkin paths.We focus on the construction by bracketed words for the convenience of computations. Factorization in algebra is an important problem. In this paper, we obtain a unique factorization of the basis elements in the free Rota-Baxter algebras of bracketed words.