Some notes on graded weakly 1-absorbing primary ideals

: A proper graded ideal P of a commutative graded ring R is called graded weakly 1-absorbing primary if whenever x y z , , are nonunit homogeneous elements of R with ≠ ∈ xyz P 0 , then either ∈ xy P or z is in the graded radical of P . In this article, we explore more results on graded weakly 1-absorbing primary ideals.


Introduction
In dispersion through this article, G is a group and R is a commutative ring with nonzero unity 1 unless specified differently.
, where R g is an additive subgroup of R for all ∈ g G, then R is aforementioned to be a graded ring (gr-R).The aspects of R g are called homogeneous of degree g.If ∈ s R, then s can be expressed uniquely as ∑ ∈ s g G g , where s g is the component of s in R g , and = s 0 g is represented by the symbol.The set of all homogeneous aspects of R is ⋃ ∈ R g G g and is denoted by ( ) h R .The component R e is a subring of R and ∈ R 1 e .Let R be a gr-R and P be an ideal of R.Then, P is aforementioned to be a graded ideal (gr-I) if ( ) = ⊕ ⋂ ∈ P P R g G g , i.e., for ∈ p P, ∈ p P g for all ∈ g G.An ideal of a gr-R is not necessarily gr-I.For a G-gr-R R and a gr-I P of R, ∕ R P is a G-gr-R with ( ) ( ) ∕ = + ∕ R P R P P g g for all ∈ g G.For further phrasing, see [1].
A proper gr-I P of R is aforementioned to be a graded prime ideal (gr-p-I) if ∈ xy P implies either ∈ x P or ∈ y P, for all ( ) ∈ x y h R , [2].It is clear that if P is a prime ideal of R and it is a gr-I, then P is a gr-p-I of R. Indeed, the example below demonstrates that a gr-p-I is not necessarily a prime ideal: Consider the gr-I = P pR of R, where p is a prime number with = + p c d Allow for P to be a gr-I of R.Then, the graded radical of P is denoted by ( ) -P Gr rad and is defined as follows: Recall that Gr-rad(P) is every time a gr-I of R [2].A proper gr-I P of R is aforementioned to be a graded primary ideal (gr-py-I) if is a gr-p-I of R and P is allegedly graded Q-primary.
Since gr-p-I's and gr-py-I's are vital in commutative graded ring theory, numerous authors have looked into various generalizations of these gr-Is.Atani [4] proposed the idea of graded weakly prime ideals.A proper gr-I P of R is called a graded weakly prime ideal (gr-wp-I) whenever ( ) ∈ x y h R , and ≠ ∈ xy P 0 , then ∈ x P or ∈ y P. Atani [5] presented the impression of graded weakly primary ideals.A proper gr-I P of R is called a graded weakly primary ideal (gr-w-py-I) of R if whenever ( ) ∈ x y h R , and ≠ ∈ xy P 0 , then ∈ x P or ( ) ∈y P Gr rad .New generalizations of graded primary ideals and graded weakly primary ideals are, accord- ingly, the notions of graded 1-absorbing primary ideals and graded weakly 1-absorbing primary ideals proposed by Abu-Dawwas and Bataineh [6,7].A proper gr-I P of R is called a graded 1-absorbing primary ideal (gr- , , where K is a field, and = G .Then, R is gr-R by , and it is obvious that P is a gr-1-ab-py-I of R. On the contrary, P is not gr-py-I of R Example 2.11 in the study by Soheilnia and Darani [8].
Definition 1.4.[6,9] Let R be a G-graded ring and P be a graded ideal of R. Assume that ∈ g G, where ≠ P R g g .Then, • P is supposedly a g-1-absorbing primary ideal (g -1-ab-py-I) of R if whenever nonunit elements • P is presumably a g-weakly 1-absorbing primary ideal (g-w-1-ab-py-I) of R, whenever nonunit elements , where ∈ xy P, then either ∈ x P or ∈ y P. • P is presumably a g-primary ideal (g -py-I) of R, if whenever ∈ x y R , g , where ∈ xy P, then either ∈ x P or ( ) ∈y P Gr rad .
• P is supposedly a g -weakly primary ideal (g-w-py-I) of R if whenever ∈ x y R , g , where ≠ ∈ xy P 0 , then either ∈ x P or ( ) ∈y P Gr rad .
In this article, we explore more outcomes on graded weakly 1-absorbing primary ideals.In fact, the study by Almahdi et al. [10] inspired quite a few of the outcomes.Among a number of outcomes, we proved that if R e is a nonlocal ring and P is an e-w-1-ab-py-I of R that is not an e-w-py-I, then either = P 0 with s as an idempotent such that ⟨ ⟩ − s 1 is a maximal ideal of R e (Theorem 2.4).In addition, we showed that if every nonzero gr-py-I of R is a gr-p-I and ( ) -Gr rad 0 is a gr-m-I of R, then either ( ) -= Gr rad 0 0 or ( ) -Gr rad 0 is the unique nonzero proper gr-I of R (Proposition 2.8).In addition, we proved that if R is a HUN-ring and { } 0 is a gr-py-I of R, then R is a gr-loc-R with gr-m-I ( ) -Gr rad 0 (Theorem 2.12).Moreover, a nice characterization was introduced in Theorem 2.13.In addition, we showed that if R is a finitely generated gr-loc-R with gr-m-I X , R is a gr-D, and every gr-1-ab-py-I of R is a gr-w-py-I, then R is either HUN-ring or X is the unique nonzero gr-p-I of R (Theorem 2.14).Furthermore, we proved that if R is a first strongly gr-R, then every e-w-1-ab-py-I of R is an e-s-py-I if and only if ( ) -Gr rad 0 is an e-p-I of R (Proposition 2.16).Finally, we showed that if R is a reduced first strongly gr-R, then every e-w-1-ab-py-I of R is an e-1-ab-py-I if and only if R e is a domain (Proposition 2.19).

Results
Our results are presented in this paragraph.
Proposition 2.1.Let P be a gr-I of R such that ( ) Let R be a gr-R.Then, R e holds all homogeneous idempotent elements of R.
Proof.Let ( ) ∈ x h R be an idempotent.Then, ∈ x R g for some ∈ g G, and then i.e., = g e.As a deduction, ∈ x R e .□ Theorem 2.4.Allow for R to be a gr-R such that R e is a nonlocal ring.If P is an e-w-1-ab-py-I of R that is not an e- w-py-I, then with s as an idempotent such that ⟨ ⟩ − s 1 is a maximal ideal of R e .
Some notes on graded weakly 1-absorbing primary ideals  3 Proof.Let us say that (2) is not met.Since P is not an e-w-py-I, there exists , a contradiction since P is not an e-w-py-I.Thus, neither is a product of two nonfield rings.By Theorem 13 of the study by Badawi and Yetkin [12], P is an e-py-I, a contradiction.Subsequently, + v p is a nonunit, and so ( ) .Suppose that there exist ∈ p q P , e where ≠ ypq 0. Consequently, ( = + + ∈ ypq v p x q y P 0 .As above, + v p and + x q are nonunits.Hence, . Let ∈ p q t P , , e in such a way that ≠ pqt 0. Afterward, ( + + = ≠ v p x q y t pqt 0. As above, + v p, + x q, and + y t are nonunits.Then, ( )( ) + + ∈ v p x q P or ( ) + ∈y t P Gr rad .That is, ∈ vx P or ( ) ∈y P Gr rad , a contradiction.Hence, : 0 g [13].Undoubtedly, if R is strongly graded, then R is first strongly graded.The following example, however, demonstrates that the converse is not always true.
Example 2.5.Let ( ) = R M K 2 (the ring of all × 2 2 matrices with entries from a field K ) and = G 4 .Then R is gr-R by supp , .
Theorem 2.6.Let R be a first strongly gr-R such that R e is a nonlocal reduced ring.Suppose that P is an e-w-1-ab- py-I of R. If P is not an e-w-py-I, then ( ) -= P P Gr rad e e . Proof.
, then = P 0 e and = P g ( ) with s as an idempotent with the isomorphism A proper gr-I X of R is allegedly a graded maximal ideal (gr-m-I) of R if whenever I is a gr-I of R with ⊆ ⊆ X I R, then = I X or = I R. Assuredly, every gr-m-I is a gr-p-I.A gr-R R is assumed to be a graded local ring (gr-loc-R) if R has a unique gr-m-I.
Proposition 2.7.Allow for R to be a gr-loc-R with gr-m-I X. Assume that P is a gr-p-I of R such that ⊆ P X.Then, PX is a gr-1-ab-py-I of R.
Proof. .Thus, PX is a gr-1-ab-py-I of R. □ Proposition 2.8.Allow for R to be a gr-R such that every nonzero gr-py-I of R is a gr-p-I.If ( ) -Gr rad 0 is a gr-m-I of R, then either ( ) -= Gr rad 0 0 or ( ) -Gr rad 0 is the unique nonzero proper gr-I of R.
Proof.If R is a gr-D, then ( ) -= Gr rad 0 0. Assume that R is not a gr-D.Allow for J to be a nonzero proper gr-I of R.Then, ( ) ( ) -⊆ -J Gr rad 0 Gr rad , and then as ( ) -Gr rad 0 is a gr-m-I of R, ( ) ( ) -= -J Gr rad 0 Gr rad .So, ( ) -J Gr rad is a gr-m-I of R, which implies that J is a gr-py-I of R by Proposition 1.11 of the study by Refai and Al-Zoubi [3], and then J is a gr-p-I of R, and so ( ) ( ) = -= -J J Gr rad Gr rad 0 .As a consequence, ( ) -Gr rad 0 is the unique nonzero proper gr-I of R. □ A gr-R R is said to be a graded domain (gr-D) if R has no homogeneous zero divisors, and is said to be a graded field (gr-F) if every nonzero homogeneous element of R is unit [1].Assuredly, if R is a domain (field) and it is graded, then R is a gr-D (gr-F).Nevertheless, Example 2.4 of the study by Abu-Dawwas [14] shows that a gr- D (gr-F) is not necessarily a domain (field).Recall from [15] and [16,Proposition 2.25], if every element of R is either nilpotent or unit, or alternatively if all of its nonunit elements are products of unit and nilpotent elements, then R is said to be a UN-ring.A straightforward UN-ring example is ∕9 .In fact, we present the idea of HUN-rings: Definition 2.9.A gr-R R is presumably a HUN-ring if every homogeneous element of R is either a unit or a nilpotent.
i.e., p divides xb in , and again p divides x or p divides b, which implies that p divides x or p divides = y ib in R.
So, P is a gr-p-I of R. On the other hand, P is not a prime ideal of R since ( [11]v u is a unit.Thus, by Lemma 1 in the study by Badawi and Celikel[11], R e is a local ring, a contradiction.Because of that, there exists a nonunit ∈ without any doubt, K is a field and e e since R 1 is reduced.□