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Zbl 0979.13003
Smith, Patrick F.
Primary modules over commutative rings. (English)
[J] Glasg. Math. J. 43, No.1, 103-111 (2001). ISSN 0017-0895; ISSN 1469-509X/e

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 $N\ne M$ 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 $\text {rad}_M(0)$.\par 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 $M$.\par 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.
[Iuliu Crivei (Cluj-Napoca)]

MSC 2000:: *13A10 Radical theory on commutative rings
13C05 Structure of modules (commutative rings)
13A15 Ideals; multiplicative ideal theory

Keywords: radical formula for primary submodules

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