## 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