## Inert extensions of Krull domains

HTML articles powered by AMS MathViewer

- by Douglas L. Costa and Jon L. Johnson PDF
- Proc. Amer. Math. Soc.
**59**(1976), 189-194 Request permission

## Abstract:

Let $A \subseteq B$ be integral domains with $B$ an inert extension of a Krull domain $A$. Let $\mathcal {P}(A)$ be the set of height one primes of $A$, and let $T = { \cap _{p \in \mathcal {P}(A)}}B \otimes {A_p}$. When each ${B_p} = B \otimes {A_p}$ is a UFD, a necessary and sufficient condition for $T$ to be a Krull domain is obtained. If $T$ is a Krull domain and each ${B_p}$ is a UFD, then the divisor class groups of $A$ and $T$ are isomorphic under the natural mapping. These results are applied to $A \subseteq B$ when $B$ is a symmetric algebra over $A$ and when $B$ is locally a polynomial ring over $A$.## References

- M. F. Atiyah and I. G. Macdonald,
*Introduction to commutative algebra*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR**0242802** - Jacob Barshay,
*Graded algebras of powers of ideals generated by $A$-sequences*, J. Algebra**25**(1973), 90â99. MR**332748**, DOI 10.1016/0021-8693(73)90076-8 - P. M. Cohn,
*Bezout rings and their subrings*, Proc. Cambridge Philos. Soc.**64**(1968), 251â264. MR**222065**, DOI 10.1017/s0305004100042791
D. L. Costa, - Douglas L. Costa,
*Unique factorization in modules and symmetric algebras*, Trans. Amer. Math. Soc.**224**(1976), no.Â 2, 267â280 (1977). MR**422250**, DOI 10.1090/S0002-9947-1976-0422250-1 - Paul Eakin and James Silver,
*Rings which are almost polynomial rings*, Trans. Amer. Math. Soc.**174**(1972), 425â449. MR**309924**, DOI 10.1090/S0002-9947-1972-0309924-4 - Robert M. Fossum,
*The divisor class group of a Krull domain*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74, Springer-Verlag, New York-Heidelberg, 1973. MR**0382254** - Robert Gilmer,
*Multiplicative ideal theory*, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972. MR**0427289** - Irving Kaplansky,
*Commutative rings*, Allyn and Bacon, Inc., Boston, Mass., 1970. MR**0254021** - A. Micali, P. Salmon, and P. Samuel,
*IntĂ©gritĂ© et factorialitĂ© des algĂšbres symĂ©triques*, Proc. Fourth Brazilian Math. Colloq. (1963) (Portuguese), Conselho Nacional de Pesquisas, SĂŁo Paulo, 1965, pp.Â 61â76 (French). MR**0207741** - Anne-Marie Nicolas,
*Modules factoriels*, Bull. Sci. Math. (2)**95**(1971), 33â52 (French). MR**284426** - Anne-Marie Nicolas,
*Extensions factorielles et modules factorables*, Bull. Sci. Math. (2)**98**(1974), no.Â 2, 117â143 (French). MR**424788** - Pierre Samuel,
*Anneaux graduĂ©s factoriels et modules rĂ©flexifs*, Bull. Soc. Math. France**92**(1964), 237â249 (French). MR**186702** - P. Samuel,
*Lectures on unique factorization domains*, Tata Institute of Fundamental Research Lectures on Mathematics, No. 30, Tata Institute of Fundamental Research, Bombay, 1964. Notes by M. Pavman Murthy. MR**0214579**

*Symmetric algebras and retracts*, Dissertation, Univ. of Kansas, 1974.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**59**(1976), 189-194 - MSC: Primary 13F05
- DOI: https://doi.org/10.1090/S0002-9939-1976-0412173-1
- MathSciNet review: 0412173