Geodesic
On valuation subalgebras and their centres
Glasgow mathematical journal, Tome 31 (1989) no. 1, pp. 115-126

Voir la notice de l'article provenant de la source Cambridge University Press

We shall extend results of Samuel [19] and Griffin [8, 9] about conditions which generalise the notion of valuation domain in a field. Let U be a commutative ring with identity, R a subring of U and L an R-submodule of U. The conditions we study have in common the property (EV), that the submodules L:x (x ∈ U) form a chain. We pay particular attention to the strongest of the conditions, viz, that L be a Manis valuation (MV) subring, i.e. having a prime ideal P such that (L, P) is a maximal pair in U (see [19], [16] and e.g. [4]). Such P is unique, being the union of all L:x such that x ∉ L, which we call P+(L) the centre of L. This set P+ plays a key role in the study of all our valuation conditions. 
Rhodes, C. P. L. On valuation subalgebras and their centres. Glasgow mathematical journal, Tome 31 (1989) no. 1, pp. 115-126. doi: 10.1017/S0017089500007606
@article{10_1017_S0017089500007606,
     author = {Rhodes, C. P. L.},
     title = {On valuation subalgebras and their centres},
     journal = {Glasgow mathematical journal},
     pages = {115--126},
     year = {1989},
     volume = {31},
     number = {1},
     doi = {10.1017/S0017089500007606},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007606/}
}
TY  - JOUR
AU  - Rhodes, C. P. L.
TI  - On valuation subalgebras and their centres
JO  - Glasgow mathematical journal
PY  - 1989
SP  - 115
EP  - 126
VL  - 31
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007606/
DO  - 10.1017/S0017089500007606
ID  - 10_1017_S0017089500007606
ER  - 
%0 Journal Article
%A Rhodes, C. P. L.
%T On valuation subalgebras and their centres
%J Glasgow mathematical journal
%D 1989
%P 115-126
%V 31
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089500007606/
%R 10.1017/S0017089500007606
%F 10_1017_S0017089500007606

[1] 1.Bourbaki, N., Commutative algebra (Addison-Wesley, 1972). Google Scholar

[2] 2.Brase, C. H., Valuation ideals with zero divisors, J. Reine Angew. Math. 258 (1973), 192–200. Google Scholar

[3] 3.Eggert, N. and Rutherford, H., A local characterization of Prüfer rings, J. Reine Angew. Math. 250 (1971), 109–112. Google Scholar

[4] 4.Faith, C., The structure of valuation rings, j. Pure Appl. Algebra 31 (1984), 7–27. Google Scholar | DOI

[5] 5.Faith, C., The structure of valuation rings II, J. Pure Appl. Algebra 42 (1986), 37–43. Google Scholar | DOI

[6] 6.Gilmer, R., Multiplicative ideal theory (Marcel Dekker, 1972). Google Scholar

[7] 7.Griffin, M., Prüfer rings with zero divisors, J. Reine Angew. Math. 239/240 (1969), 55–67. Google Scholar

[8] 8.Griffin, M., Generalising valuations to commutative rings, Queen's Mathematical Preprint No. 1970–40 (Queen's University, Kingston, Ontario, 1970). Google Scholar

[9] 9.Griffin, M., Valuations and Prüfer rings, Canad. J. Math. 26 (1974), 412–429. Google Scholar | DOI

[10] 10.Harrison, D. K., Finite and infinite primes for rings and fields (Mem. Amer. Math. Soc. 68), (1966). Google Scholar

[11] 11.Huckaba, J. A., On valuation rings that contain zero divisors, Proc. Amer. Math. Soc. 40 (1973), 9–15. Google Scholar | DOI

[12] 12.Kirby, D., Integral dependence and valuation algebras, Proc. London Math. Soc. (3) 20 (1970), 79–100. Google Scholar | DOI

[13] 13.Kirby, D. and Mehran, H. A., Homomorphisms and the space of valuation modules, Quart. J. Math. Oxford Ser. (2) 21 (1970), 439–443. Google Scholar | DOI

[14] 14.Kirby, D. and Mehran, H. A., Fractional powers of a submodule of an algebra, Mathematika 18 (1971), 8–13. Google Scholar | DOI

[15] 15.Larsen, M. D. and McCarthy, P. J., Multiplicative theory of ideals (Academic Press, New York, 1971). Google Scholar

[16] 16.Manis, M. E., Valuations on a commutative ring, Proc. Amer. Math. Soc. 20 (1969), 193–198. Google Scholar | DOI

[17] 17.Rand, D. A., Non-unitary valuation subalgebras, Proc. London Math. Soc. (3) 29 (1974), 485–501. Google Scholar | DOI

[18] 18.Rhodes, C. P. L., Valuation subalgebras and analogues of the decomposition of a radical ideal into prime ideals, Proc. London Math. Soc. (3) 46 (1983), 385–410. Google Scholar | DOI

[19] 19.Samuel, P., La notion de place dans un anneau, Bull. Soc. Math. France 85 (1957), 123–133. Google Scholar | DOI

[20] 20.van Geel, J., Places and valuations in noncommutative ring theory (Marcel Dekker, 1981). Google Scholar

Cité par Sources :