Definition. A commutative ring $R$ with non-zero unity is a *Prüfer ring* (cf. Prüfer domain) if every finitely generated regular ideal of $R$ is invertible. (It can be proved that if every regular ideal of $R$ generated by two elements is invertible, then all finitely generated regular ideals are invertible; cf. invertibility of regularly generated ideal.)

Denote generally by $\mathfrak{m}_{p}$ the $R$-module generated by the coefficients of a polynomial $p$ in $T[x]$, where $T$ is the total ring of fractions of $R$. Such coefficient modules are, of course, fractional ideals of $R$.

Theorem 1 (Pahikkala 1982). Let $R$ be a commutative ring with non-zero unity and let $T$ be the total ring of fractions of $R$. Then, $R$ is a Prüfer ring iff the equation

$\displaystyle\mathfrak{m}_{f}\mathfrak{m}_{g}=\mathfrak{m}_{fg}$ | (1) |

holds whenever $f$ and $g$ belong to the polynomial ring $T[x]$ and at least one of the fractional ideals $\mathfrak{m}_{f}$ and $\mathfrak{m}_{g}$ is regular. (See also product of finitely generated ideals.)

Theorem 2 (Pahikkala 1982). The commutative ring $R$ with non-zero unity is Prüfer ring iff the multiplication rule

$(a,\,b)(c,\,d)=(ac,\,ad+bc,\,bd)$ |

for the integral ideals of $R$ holds whenever at least one of the generators $a$, $b$, $c$ and $d$ is not zero divisor.

The proofs are found in the paper

J. Pahikkala 1982: “Some formulae for multiplying and inverting ideals”. – Annales universitatis turkuensis 183. Turun yliopisto (University of Turku).

Cf. the entries “multiplication rule gives inverse ideal” and “two-generator property”.

An additional characterization of Prüfer ring is found here in the entry “least common multiple”, several other characterizations in [1] (p. 238–239).

Note. A commutative ring $R$ satisfying the equation (1) for all polynomials $f,\,g$ is called a Gaussian ring. Thus any Prüfer domain is always a Gaussian ring, and conversely, an integral domain, which is a Gaussian ring, is a Prüfer domain. Cf. [2].

## References

- 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press. New York (1971).
- 2 Sarah Glaz: “The weak dimensions of Gaussian rings”. – Proc. Amer. Math. Soc. (2005).

## Comments

## Lowew index

Please, tell me, how I can do the lower index "fg"

in my object Â«PrÃ¼fer ringÂ» on the right hand side of the first formula?

That is, there should be "M" with the product "fg" as lower index.

I have tried the form M_f_g, but it gives the error message "double index". Now I have written M_fg, but this does not give the right result.

pahio

## Re: Lowew index

M_{fg}

## Re: Lower index

Thank you very much!