Primary modules over commutative rings (Q5890199)

Let \(R\) be a commutative ring. All modules considered are unital \(R\)-modules. For an ideal \(I\) of \(R\) and for a submodule \(N\) of an \(R\)-module \(M\) the following sets are defined: NEWLINE\[NEWLINE\sqrt I=\{r\in R:r^n\in I\text{ for some positive integer }n\},NEWLINE\]NEWLINE NEWLINE\[NEWLINE(N:M)= \{r\in R:rM \subseteq N\},NEWLINE\]NEWLINE NEWLINE\[NEWLINEE_M(N)= \{rm:r \in R,\;m\in M \text{ and }r^km\in N\text{ for some positive integer }k.NEWLINE\]NEWLINE By \(RE_M(N)\) will be denoted the submodule of \(M\) generated by the non-empty subset \(E_M(N)\) of \(M\). -- A submodule \(N\) of \(M\) is called prime (respectively, primary) if \(N\neq M\) and whenever \(r\in R\), \(m\in M\) and \(rm\in N\) then \(m\in N\) or \(r\in(N:M)\) (respectively, \(r\in\sqrt{(N:M)})\). The module \(M\) will be called primary if its zero submodule is primary. For any submodule \(N\) of an \(R\)-module \(M\), the radical, \(\text{rad}_M(N)\), of \(N\) is defined to be the intersection of all prime submodule of \(M\) containing \(N\) and \(\text{rad}_M(N)=M\) if \(N\) is not contained in any prime submodules of \(M\). The radical of the module \(M\) is defined to be \(\text{rad}_M(0)\).NEWLINENEWLINENEWLINEThe author gives the definition that the module \(M\) satisfies the radical formula for primary submodules if \(\text{rad}_M(N)= RE_M(N)\) for every primary submodule \(N\) of \(M\).NEWLINENEWLINENEWLINEThe main result is: If \(R\) is a commutative domain which is either Noetherian or a UFD then \(R\) is one-dimensional if and only if every (finitely generated) primary \(R\)-module has prime radical, and this holds precisely when every (finitely generated) \(R\)-module satisfies the radical formula for primary submodules.