Primary modules over commutative rings. (English) Zbl 0979.13003

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: \[ \sqrt I=\{r\in R:r^n\in I\text{ for some positive integer }n\}, \]
\[ (N:M)= \{r\in R:rM \subseteq N\}, \]
\[ E_M(N)= \{rm:r \in R,\;m\in M \text{ and }r^km\in N\text{ for some positive integer }k. \] 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 \(\text{rad}_M(0)\).
The 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 \(M\).
The 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.


13A10 Radical theory on commutative rings (MSC2000)
13C05 Structure, classification theorems for modules and ideals in commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
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