[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/BKMS.b200962

ON GRADED RADICALLY PRINCIPAL IDEALS

Abu-Dawwas, Rashid (Department of Mathematics Yarmouk University)

Publication Information

Bulletin of the Korean Mathematical Society / v.58, no.6, 2021 , pp. 1401-1407 More about this Journal

Abstract

Let R be a commutative G-graded ring with a nonzero unity. In this article, we introduce the concept of graded radically principal ideals. A graded ideal I of R is said to be graded radically principal if Grad(I) = Grad(〈c〉) for some homogeneous c ∈ R, where Grad(I) is the graded radical of I. The graded ring R is said to be graded radically principal if every graded ideal of R is graded radically principal. We study graded radically principal rings. We prove an analogue of the Cohen theorem, in the graded case, precisely, a graded ring is graded radically principal if and only if every graded prime ideal is graded radically principal. Finally we study the graded radically principal property for the polynomial ring R[X].

Keywords

Graded radical ideals; graded principal ideals; graded radically principal ideals; graded radically principal rings;

Citations & Related Records

Reference

1	W. T. Ashby, On graded principal ideal domains, JP J. Algebra Number Theory Appl. 24 (2012), no. 2, 159-171.
2	S. E. Atani and U. Tekir, On the graded primary avoidance theorem, Chiang Mai J. Sci. 34 (2007), no. 2, 161-164.
3	R. Hazrat, Graded rings and graded Grothendieck groups, London Mathematical Society Lecture Note Series, 435, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316717134
4	C. Nastasescu and F. Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, 1836, Springer-Verlag, Berlin, 2004. https://doi.org/10.1007/b94904
5	M. Refai and R. Abu-Dawwas, On generalizations of graded second submodules, Proyecciones 39 (2020), no. 6, 1537-1554. DOI
6	M. Refai, M. Hailat, and S. Obiedat, Graded radicals and graded prime spectra, Far East J. Math. Sci. (FJMS) 2000, Special Volume, Part I, 59-73.
7	M. Refai, Various types of strongly graded rings, Abhath Al-Yarmouk Journal (Pure Sciences and Engineering Series) 4 (1995), no. 2, 9-19.
8	M. Aqalmoun and M. El Ouarrachi, Radically principal rings, Khayyam J. Math. 6 (2020), no. 2, 243-249.
9	F. Farzalipour and P. Ghiasvand, On the union of graded prime submodules, Thai J. Math. 9 (2011), no. 1, 49-55.