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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.

MSC:

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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