HOME JOURNALS CONTACT

Journal of Applied Sciences

Year: 2003 | Volume: 3 | Issue: 2 | Page No.: 71-75
DOI: 10.3923/jas.2003.71.75
Principal Ideal Rings
Ahmet Goksel Agargun

Abstract: In this paper we continue to extend ring concepts. Here we define principal ideal rings for commutative rings (not necessarily with identity) and prove that this definition is equivalent to the usual definition in the case of a ring with identity. Then we generalize some results for principal ideal rings. We study direct sums, direct summands and quotient rings. We show that every Euclidean ring is a principal ideal ring.

Fulltext PDF

How to cite this article
Ahmet Goksel Agargun , 2003. Principal Ideal Rings. Journal of Applied Sciences, 3: 71-75.

Keywords: Euclidean rings, euclidean algorithms, principal ideal and principal ideal ring

REFERENCES
  • Agargun, A.G. and C.R. Fletcher, 1995. Euclidean rings. Turk. J. Math., 19: 291-299.

  • Agargum, A.G., 1997. On euclidean rings proyecciones. Revista Matematica, 16: 23-36.

  • Agargum, A.G. and B.A. Ersoy, 2000. About euclidean ring. YTU Dergisi, pp: 36-44.

  • Amano, K., 1985. A note on euclidean ring. Bull. Fac. Gen. Gifu Univ., No. 20, pp: 13-15.

  • Fletcher, C.R., 1971. Euclidean rings. J. Lond. Math. Soc., 2: 79-82.

  • Hibolt, J.J., 1977. Correction Une note sur les anneaux euclidiens. Comptes Rendues, 284: 847-847.

  • Hibolt, J.J., 1975. Des anneaus euclidines don`t le plus petit algorithme n`est pas valeurs finies. Comptes Rendues, 281: 411-414.

  • Hungerford, T.W., 1974. Algebra, Graduate Texts in Mathematics. Springer Verlag, New York

  • Kanemitsu, M. and K. Yoshida, 1986. Euclidean rings. Bull. Fac. Sci., Ibaraki Univ. Math., No. 18.

  • Motzkin, T., 1949. The euclidean algorithm. Bull. Am. Math. Soc., 55: 1142-1146.

  • Nagata, M., 1978. On euclidean algorithm. Tata Inst. Fund. Res. Stud. Math., 8: 175-186.

  • Nagata, M., 1985. Some remarks on euclid rings. J. Math. Kyoto Univ., 25: 421-422.

  • Nagata, M., 1987. On the definition of euclidean ring. Adv. Stud. Pure Math., 11: 167-171.

  • Samuel, P., 1971. About euclidean rings. J. Algebra, 19: 282-301.

    • © Science Alert. All Rights Reserved