Properties of ϕ-Primal Graded Ideals
Let R be a commutative graded ring with unity 1≠0. A proper graded ideal of R is a graded ideal I of R such that I≠R. Let ϕ:I(R)→I(R)∪{∅} be any function, where I(R) denotes the set of all proper graded ideals of R. A homogeneous element a∈R is ϕ-prime to I if ra∈I-ϕ(I) where r is a homogeneous elem...
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Wiley
2017-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2017/3817479 |
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| author | Ameer Jaber |
| author_facet | Ameer Jaber |
| author_sort | Ameer Jaber |
| collection | DOAJ |
| description | Let R be a commutative graded ring with unity 1≠0. A proper graded ideal of R is a graded ideal I of R such that I≠R. Let ϕ:I(R)→I(R)∪{∅} be any function, where I(R) denotes the set of all proper graded ideals of R. A homogeneous element a∈R is ϕ-prime to I if ra∈I-ϕ(I) where r is a homogeneous element in R; then r∈I. An element a=∑g∈Gag∈R is ϕ-prime to I if at least one component ag of a is ϕ-prime to I. Therefore, a=∑g∈Gag∈R is not ϕ-prime to I if each component ag of a is not ϕ-prime to I. We denote by νϕ(I) the set of all elements in R that are not ϕ-prime to I. We define I to be ϕ-primal if the set P=νϕ(I)+ϕ(I) (if ϕ≠ϕ∅) or P=νϕ(I) (if ϕ=ϕ∅) forms a graded ideal of R. In the work by Jaber, 2016, the author studied the generalization of primal superideals over a commutative super-ring R with unity. In this paper we generalize the work by Jaber, 2016, to the graded case and we study more properties about this generalization. |
| format | Article |
| id | doaj-art-464c9a9f04344555a062b81677c6d2c6 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2017-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-464c9a9f04344555a062b81677c6d2c62025-02-03T05:54:08ZengWileyJournal of Mathematics2314-46292314-47852017-01-01201710.1155/2017/38174793817479Properties of ϕ-Primal Graded IdealsAmeer Jaber0Department of Mathematics, The Hashemite University, Zarqa 13115, JordanLet R be a commutative graded ring with unity 1≠0. A proper graded ideal of R is a graded ideal I of R such that I≠R. Let ϕ:I(R)→I(R)∪{∅} be any function, where I(R) denotes the set of all proper graded ideals of R. A homogeneous element a∈R is ϕ-prime to I if ra∈I-ϕ(I) where r is a homogeneous element in R; then r∈I. An element a=∑g∈Gag∈R is ϕ-prime to I if at least one component ag of a is ϕ-prime to I. Therefore, a=∑g∈Gag∈R is not ϕ-prime to I if each component ag of a is not ϕ-prime to I. We denote by νϕ(I) the set of all elements in R that are not ϕ-prime to I. We define I to be ϕ-primal if the set P=νϕ(I)+ϕ(I) (if ϕ≠ϕ∅) or P=νϕ(I) (if ϕ=ϕ∅) forms a graded ideal of R. In the work by Jaber, 2016, the author studied the generalization of primal superideals over a commutative super-ring R with unity. In this paper we generalize the work by Jaber, 2016, to the graded case and we study more properties about this generalization.http://dx.doi.org/10.1155/2017/3817479 |
| spellingShingle | Ameer Jaber Properties of ϕ-Primal Graded Ideals Journal of Mathematics |
| title | Properties of ϕ-Primal Graded Ideals |
| title_full | Properties of ϕ-Primal Graded Ideals |
| title_fullStr | Properties of ϕ-Primal Graded Ideals |
| title_full_unstemmed | Properties of ϕ-Primal Graded Ideals |
| title_short | Properties of ϕ-Primal Graded Ideals |
| title_sort | properties of ϕ primal graded ideals |
| url | http://dx.doi.org/10.1155/2017/3817479 |
| work_keys_str_mv | AT ameerjaber propertiesofphprimalgradedideals |