## Rocky Mountain Journal of Mathematics

### Weakly factorial property of a generalized Rees ring $D[X,d/X]$

Gyu Whan Chang

#### Abstract

Let $D$ be an integral domain, $X$ an indeterminate over $D$, $d \in D$, and $R = D[X, {d}/{X}]$ a subring of $D[X,{1}/{X}]$. In this paper, we show that $R$ is a weakly factorial domain if and only if $D$ is a weakly factorial GCD-domain and $d=0$, $d$ is a unit of $D$ or $d$ is a prime element of $D$. We also show that, if $D$ is a weakly factorial GCD-domain, $p$ is a prime element of $D$, and $n \geq 2$ is an integer, then $D[X, {p^n}/{X}]$ is an almost weakly factorial domain with $Cl(D[X, {p^n}/{X}]) = \mathbb {Z}_n$.

#### Article information

Source
Rocky Mountain J. Math., Volume 48, Number 7 (2018), 2175-2185.

Dates
First available in Project Euclid: 14 December 2018

https://projecteuclid.org/euclid.rmjm/1544756806

Digital Object Identifier
doi:10.1216/RMJ-2018-48-7-2175

Mathematical Reviews number (MathSciNet)
MR3892129

Zentralblatt MATH identifier
06999259

#### Citation

Chang, Gyu Whan. Weakly factorial property of a generalized Rees ring $D[X,d/X]$. Rocky Mountain J. Math. 48 (2018), no. 7, 2175--2185. doi:10.1216/RMJ-2018-48-7-2175. https://projecteuclid.org/euclid.rmjm/1544756806

#### References

• D.D. Anderson and D.F. Anderson, The ring $R[X, r/X]$, Lect. Notes Pure Appl. Math. 171 (1995), 95–113.
• D.D. Anderson, D.F. Anderson and M. Zafrullah, Splitting the $t$-class group, J. Pure Appl. Alg. 74 (1991), 17–37.
• D.D. Anderson, E. Houston and M. Zafrullah, $t$-linked extensions, the $t$-class group, and Nagata's theorem, J. Pure Appl. Alg. 86 (1993), 109–124.
• D.D. Anderson, J.L. Mott and M. Zafrullah, Finite character representations for integral domains, Boll. Un. Mat. 6 (1992), 613–630.
• D.D. Anderson and M. Zafrullah, Weakly factorial domains and groups of divisiblity, Proc. Amer. Math. Soc. 109 (1990), 907–913.
• ––––, Almost Bézout domains, J. Algebra 142 (1991), 141–309.
• D.F. Anderson and G.W. Chang, Homogeneous splitiing sets of a graded integral domain, J. Algebra 288 (2005), 527–544.
• D.F. Anderson, G.W. Chang and J. Park, Generalized weakly factorial domains, Houston J. Math. 29 (2003), 1–13.
• A. Bouvier, Le groupe des classes d'un anneau intégré, Natl. Soc. Savantes, Brest 4 (1982), 85–92.
• G.W. Chang, Semigroup rings as weakly factorial domains, Comm. Alg. 37 (2009), 3278–3287.
• R. Fossum, The divisor class group of a Krull domain, Springer, New York, 1973.
• R. Gilmer, Multiplicative ideal theory, Marcel Dekker, New York, 1972.
• M. Griffin, Some results on $v$-multiplication rings, Canadian Math. J. 19 (1967), 710–722.
• E. Houston and M. Zafrullah, On $t$-invertibility, II, Comm. Alg. 17 (1989), 1955–1969.
• H. Kim, T.I. Kwon and Y.S. Park, Factorization and divisibility in generalized Rees rings, Bull. Korean Math. Soc. 41 (2004), 473–482.
• R. Matsuda, Torsion-free abelian semigroup rings, IV, Bull. Fac. Sci. Ibaraki Univ. 10 (1978), 1–27.
• J. Mott, The group of divisibility of Rees rings, Math. Japonica 20 (1975), 85–87.
• D. Whitman, A note on unique factorization in Rees rings, Math. Japonica 17 (1972), 13–14.