Reza Ebrahimi ATANİ, Shahabaddin Ebrahimi ATANİ

Graded multiplication modules and the graded ideal θg(M)

Let G be a group and let R be a G-graded commutative ring. For a graded R-module M, the notion of the associated graded ideal θg(M)of R is deﬁned. It is proved that the graded ideal θg(M)isimportant in the study of graded multiplication modules. Among various application given, the following results are proved: if M is a graded faithful multiplication module, then θg(M) is an idempotent graded multiplication ideal of R such that θg(θg(M)) = θg(M) , and every graded representable multiplication R-module is ﬁnitely generated.

PDF

___

[1] Anderson, D. D.: Some remarks on multiplication ideals, Math. Japan, 25, 463-469, (1980).
[2] Anderson, D. D. and Al-Shaniaﬁ, Y.: Multiplication modules and the ideal θ(M) , Comm. Algebra, 30, 3383- 3390, (2002).
[3] Barnard, A.: Multiplication modules, J. Algebra, 71, 174-178, (1981).
[4] Escoriza, J. and Torrecillas, B.: Multiplication graded rings, Lecture Notes in Pure and Applied Mathematics, 208, 127-137, (2000).
[5] Escoriza, J. and Torrecillas, B.: Multiplication Objects in Commutative Grothendieck Categories, Comm. Algebra, 26 (6), 1867-1883, (1998).
[6] Elbast, Z. A. and Smith, P. F.: Multiplication modules, Comm. Algebra, 16, 755-779, (1988).
[7] Ebrahimi Atani S.: On graded weakly prime ideals, Turkish Journal of Mathematics, 30, 351-358, (2006).
[8] Ebrahimi Atani S. and Farzalipour F.: On graded secondary modules, Turkish Journal of Mathematics, 31, 371-378, (2007).
[9] Ebrahimi Atani S. and Farzalipour F.: On graded multiplication modules, submitted.
[10] Ebrahimi Atani S: Multiplication modules and related results, Archivum Mathematicum, 40, 407-414, (2004).
[11] Macdonald I. G.: Secondary representation of modules over commutative rings, Sympos. Math. XI, 23-43, (1973).
[12] Sharp R. Y: Asymptotic behavior of certain sets of attached prime ideals, J. London Math. Soc., 212-218, (1986).
[13] Nastasescu C. and Van Oystaeyen F.: Graded Rings Theory, Mathematical Library 28, North Holand, Amsterdam, (1982).