Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative ring with identity. In this paper, we characterize the prime submodules of a free $R$-module $F$ of finite rank with at most $n$ generators, when $R$ is a $\text{GCD}$ domain. Also, we show that if $R$ is a B\'ezout domain, then every prime submodule with $n$ generators is the row space of a prime matrix. Finally, we study the existence of primary decomposition of a submodule of $F$ over a B\'ezout domain and characterize the minimal primary decomposition of this submodule.

Keywords: Primary decompositions, prime submodules, free modules, GCD domains, B\'ezout domains, valuation domains

MSC numbers: 13C10, 13C99, 13G99, 13F30

Supported by: This work was financially supported by the Iran National Science Foundation: INSF.