Geodesic
Arithmetical Semigroup Rings
Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1361-1371

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

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4]. 
Hardy, Bonnie R.; Shores, Thomas S. Arithmetical Semigroup Rings. Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1361-1371. doi: 10.4153/CJM-1980-106-x
@article{10_4153_CJM_1980_106_x,
     author = {Hardy, Bonnie R. and Shores, Thomas S.},
     title = {Arithmetical {Semigroup} {Rings}},
     journal = {Canadian journal of mathematics},
     pages = {1361--1371},
     year = {1980},
     volume = {32},
     number = {6},
     doi = {10.4153/CJM-1980-106-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-106-x/}
}
TY  - JOUR
AU  - Hardy, Bonnie R.
AU  - Shores, Thomas S.
TI  - Arithmetical Semigroup Rings
JO  - Canadian journal of mathematics
PY  - 1980
SP  - 1361
EP  - 1371
VL  - 32
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-106-x/
DO  - 10.4153/CJM-1980-106-x
ID  - 10_4153_CJM_1980_106_x
ER  - 
%0 Journal Article
%A Hardy, Bonnie R.
%A Shores, Thomas S.
%T Arithmetical Semigroup Rings
%J Canadian journal of mathematics
%D 1980
%P 1361-1371
%V 32
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-106-x/
%R 10.4153/CJM-1980-106-x
%F 10_4153_CJM_1980_106_x

[1] 1. Asano, K., Über Kommutative Ringe, in dene jedes Ideal als Produkt von Primidealen darstellbar ist, J. Math. Soc. Japan 3 (1951), 82–90. Google Scholar

[2] 2. Bourbaki, N., Elements of mathematics, commutative algebra (Hermann, Paris, 1972). Google Scholar

[3] 3. Connell, I. G., On the group ring, Can. J. Math. 15 (1963), 650–685. Google Scholar

[4] 4. Chouinard, L., Hardy, B. and Shores, T., Arithmetical and semiher editor y semigroup rings, preprint. Google Scholar

[5] 5. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. I (Amer. Math. Soc, Providence, R.I., 1961). Google Scholar

[6] 6. Endo, S., On semi-hereditary rings, J. Math. Soc. Japan 13 (1961), 109–119. Google Scholar

[7] 7. Fuchs, L., Über die Ideale Arithmetischer Ringe, Comment. Math. Helv. 23 (1949), 334–341. Google Scholar

[8] 8. Fuchs, L., Infinite abelian groups (Academic Press, New York, 1970). Google Scholar

[9] 9. Griffin, M., Prüfer rings with zero divisors, J. reine angew. Math. 239/240 (1970), 55–67. Google Scholar

[10] 10. Grothendieck, A. and Dieudonné, J., Elements de géométrie algébrique I (Springer-Verlag, Berlin, 1971). Google Scholar

[11] 11. Gilmer, R. and Parker, T., Semigroup rings as Prüfer rings, Duke Math. J. 41 (1974), 219–230. Google Scholar

[12] 12. Gulliksen, T., Ribenboim, R. and Viswanathan, T. M., An elementary note on group rings, J. reine angew. Math. 242 (1970), 148–162. Google Scholar

[13] 13. Goursaud, J. M. and Valette, J., Anneaux de groupe hereditaires et semi-hereditaires, J. Algebra 34 (1975), 205–212. Google Scholar

[14] 14. Jensen, C. U., Arithmetical rings, Acta. Math. Hungr. 17 (1966), 115–123. Google Scholar

[15] 15. Kozhukhov, I. B., Chain semigroup rings, (Russian), Uspekhi Matem. Nauk. 29 (1974), 169–170. Google Scholar

[16] 16. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Mass., 1966). Google Scholar

[17] 17. Larsen, M., Lewis, W. and Shores, T., Elementary divisor rings and finitely presented modules, Trans. Amer. Math. Soc. 187 (1974), 231–248. Google Scholar

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

[19] 19. Nicholson, W. K., Local group rings, Can. Bull. Math. 15 (1972), 137–138. Google Scholar

[20] 20. Warfield, R. B., Decomposability of finitely presented modules, Proc. Amer. Math. Soc. 25 (1970), 167–172. Google Scholar

[21] 21. Zariski, O. and Samuel, P., Commutative algebra I (Princeton, Van Nostrand, 1958). Google Scholar

Cité par Sources :