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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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2017-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2017/3817479 |
| Summary: | 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. |
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| ISSN: | 2314-4629 2314-4785 |