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Graded weakly 1-absorbing primary ideals

  • Malik Bataineh EMAIL logo and Rashid Abu-Dawwas
Published/Copyright: July 19, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

Let G be a group and R be a G -graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal P of R is said to be a graded weakly 1-absorbing primary ideal of R if whenever nonunit elements x , y , z ∈ h ( R ) such that 0 ≠ x y z ∈ P , then x y ∈ P or z n ∈ P , for some n ∈ N . Several properties of graded weakly 1-absorbing primary ideals are investigated.

Keywords: graded prime ideal; graded weakly prime ideal; graded primary ideal; graded weakly primary ideal; graded 2-absorbing primary ideal; graded weakly 2-absorbing primary ideal; graded 2-absorbing ideal; graded weakly 2-absorbing ideal; graded n-absorbing ideal
MSC 2010: 13A02; 13A15; 16W50

1 Introduction

Since graded prime and graded primary ideals have essential roles in graded commutative ring theory, many authors have studied generalizations of graded prime and graded primary ideals. Atani introduced in [1] the concept of graded weakly prime ideals. A proper graded ideal P of R is said to be a graded weakly prime ideal of R if whenever x , y ∈ h ( R ) such that 0 ≠ x y ∈ P , then x ∈ P or y ∈ P . Also, Atani introduced the notion of graded weakly primary ideals which is a generalization of graded primary ideals in [2]. A proper graded ideal P of R is said to be a graded weakly primary ideal of R if whenever x , y ∈ h ( R ) such that 0 ≠ x y ∈ P , then x ∈ P or y ∈ Grad ( P ) . For distinct generalizations of graded prime ideals and graded weakly prime ideals, the frameworks of graded 2-absorbing and graded weakly 2-absorbing ideals were defined. According to [3], a proper graded ideal P of R is said to be a graded 2-absorbing (graded weakly 2-absorbing) ideal of R if whenever a , b , c ∈ h ( R ) such that a b c ∈ P ( 0 ≠ a b c ∈ P ), then either a b ∈ P or a c ∈ P or b c ∈ P . As a generalization of graded 2-absorbing and graded weakly 2-absorbing ideals, graded 2-absorbing primary and graded weakly 2-absorbing primary ideals were defined in [4]. A proper graded ideal P of R is said to be a graded 2-absorbing primary (graded weakly 2-absorbing primary) ideal of R if whenever a , b , c ∈ h ( R ) with a b c ∈ P , then a b ∈ P or a c ∈ Grad ( P ) or b c ∈ Grad ( P ) . In a recent study in [5], we call a proper graded ideal P of R a graded 1-absorbing primary ideal of R if whenever nonunit elements a , b , c ∈ h ( R ) such that a b c ∈ P , then a b ∈ P or c ∈ Grad ( P ) .

In this article, we follow [6] to introduce and study the concept of graded weakly 1-absorbing primary ideal of a graded commutative ring R . A proper graded ideal P of R is said to be a graded weakly 1-absorbing primary ideal of R if whenever nonunit elements a , b , c ∈ h ( R ) such that 0 ≠ a b c ∈ P , then a b ∈ P or c ∈ Grad ( P ) . It is recognizable that a graded 1-absorbing primary ideal of R is a graded weakly 1-absorbing primary ideal of R . However, since { 0 } is always graded weakly 1-absorbing primary, a graded weakly 1-absorbing primary ideal of R needs not be a graded 1-absorbing primary ideal of R (see Example 2.2).

Among several results, we show that if P is a graded weakly 1-absorbing primary ideal of R and Grad ( { 0 } ) = { 0 } , then Grad ( P ) is a graded weakly prime ideal of R (Proposition 2.3). We prove that if P is a graded weakly 1-absorbing primary ideal of R , then ( P : z ) is a graded weakly primary ideal of R for every nonunit z ∈ h ( R ) − P (Proposition 2.17). We show that if R = R 1 × R 2 × ⋯ × R n , where R 1 , R 2 , … , R n are graded rings, then every proper graded ideal of R is a graded weakly 1-absorbing primary ideal of R if and only if n = 2 and R 1 , R 2 are graded fields (Proposition 2.19). In Proposition 2.22, we study graded weakly 1-absorbing primary ideals over graded homomorphisms.

1.1 Motivation

Studying graded prime ideals and their generalizations is important for several reasons. First, graded prime ideals are a natural generalization of prime ideals in commutative algebra and algebraic geometry, which are fundamental concepts in mathematics. They have many applications in areas such as algebraic geometry, algebraic number theory, and commutative algebra. Additionally, graded prime ideals are related to other important mathematical concepts such as graded rings and modules, which are important in areas such as algebraic geometry and algebraic topology. Understanding these concepts can help to provide a deeper understanding of the underlying structures in these areas of mathematics. Finally, the study of graded prime ideals and their generalizations can also have practical applications in areas such as computer science and engineering. For example, the theory of graded prime ideals can be used to study error-correcting codes and cryptography. Overall, studying graded prime ideals and their generalizations is important for both theoretical and practical reasons, and can help to deepen our understanding of many areas of mathematics and its application to other fields. For recent generalizations on graded prime ideals, see [7–9]. Also, for recent applications, one can look at [10–15].

1.2 Preliminaries

Throughout this article, G will be a group with identity e and R a commutative ring with a nonzero unity 1. R is said to be G -graded if R = ⊕ g ∈ G R g with R g R h ⊆ R g h for all g , h ∈ G , where R g is an additive subgroup of R for all g ∈ G . The elements of R g are called homogeneous of degree g . If x ∈ R , then x can be written as ∑ g ∈ G x g , where x g is the component of x in R g . Also, we set h ( R ) = ⋃ g ∈ G R g . Moreover, it has been proved in [16] that R e is a subring of R and 1 ∈ R e . Let I be an ideal of a graded ring R . Then, I is said to be graded ideal if I = ⊕ g ∈ G ( I ∩ R g ) , i.e., for x ∈ I , x = ∑ g ∈ G x g where x g ∈ I for all g ∈ G . An ideal of a graded ring need not be graded. Let R be a G -graded ring and I be a graded ideal of R . Then, R ∕ I is G -graded by ( R ∕ I ) g = ( R g + I ) ∕ I for all g ∈ G . If R and S are G -graded rings, then R × S is a G -graded ring by ( R × S ) g = R g × S g for all g ∈ G .

Lemma 1.1

([17], Lemma 2.1) Let R be a G-graded ring.

  1. If I and J are graded ideals of R , then I + J , I J , and I ⋂ J are graded ideals of R .

  2. If x ∈ h ( R ) , then Rx is a graded ideal of R .

Let P be a proper graded ideal of R . Then, the graded radical of P is Grad ( P ) and is defined to be the set of all r ∈ R such that for each g ∈ G , there exists a positive integer n g satisfies r g n g ∈ P . One can see that if r ∈ h ( R ) , then r ∈ Grad ( P ) if and only if r n ∈ P for some positive integer n .

2 Graded weakly 1-absorbing primary ideals

In this section, we introduce and study the concept of graded weakly 1-absorbing primary ideals.

Definition 2.1

A proper graded ideal P of a graded ring R is said to be a graded weakly 1-absorbing primary ideal of R if whenever nonunit elements x , y , z ∈ h ( R ) such that 0 ≠ x y z ∈ P , then x y ∈ P or z ∈ Grad ( P ) .

It is recognizable that a graded 1-absorbing primary ideal of R is a graded weakly 1-absorbing primary ideal of R . However, since { 0 } is always graded weakly 1-absorbing primary, a graded weakly 1-absorbing primary ideal of R needs not be a graded 1-absorbing primary ideal of R , see the following example:

Example 2.2

Consider R = Z 6 [ i ] and G = Z 2 . Then, R is G -graded by R 0 = Z 6 and R 1 = i Z 6 . Now, P = { 0 } is a graded weakly 1-absorbing primary ideal of R . In contrast, 2 , 3 ∈ h ( R ) such that 2.2.3 ∈ P with neither 2.2 ∈ P nor 3 ∈ Grad ( P ) . Hence, P is not a graded 1-absorbing primary ideal of R .

Proposition 2.3

Let R be a graded ring and P be a graded weakly 1-absorbing primary ideal of R . If Grad ( { 0 } ) = { 0 } , then Grad ( P ) is a graded weakly prime ideal of R .

Proof

Let x , y ∈ h ( R ) such that 0 ≠ x y ∈ Grad ( P ) . We may assume that x and y are nonunit elements. Then, there exists an even positive integer k = 2 s ( s ≥ 1 ) such that ( x y ) k ∈ P . Since Grad ( { 0 } ) = { 0 } , we have that ( x y ) k ≠ 0 . So, 0 ≠ x s x s y k ∈ P . Since P is a graded weakly 1-absorbing primary ideal of R , we conclude that either x s x s = x k ∈ P or y k ∈ Grad ( P ) , which implies that x ∈ Grad ( P ) or y ∈ Grad ( P ) . Hence, Grad ( P ) is a graded weakly prime ideal of R .□

Remark 2.4

If Grad ( P ) is a graded maximal ideal of R , then by ([18], Proposition 1.11), P is a graded primary ideal of R , and hence, P is a graded 1-absorbing primary ideal of R .

Definition 2.5

Let R be a G -graded ring and P be a graded ideal of R . Assume that g ∈ G such that P g ≠ R g . Then, P is said to be a g -weakly 1-absorbing primary ideal of R if whenever nonunit elements x , y , z ∈ R g such that 0 ≠ x y z ∈ P , then x y ∈ P or z ∈ Grad ( P ) . Also, P is said to be a g -weakly primary ideal if whenever x , y ∈ R g such that 0 ≠ x y ∈ P , then either x ∈ P or y ∈ Grad ( P ) .

Proposition 2.6

Let R be a G -graded ring, g ∈ G such that R g has no zero divisors and P be a g -weakly 1-absorbing primary ideal of R . If for every nonzero p ∈ P , there exists a nonunit w ∈ R g such that w p ≠ 0 and w + u is a nonunit element of R g for some unit u ∈ R g . Then, P is a g -weakly primary ideal of R .

Proof

Let x , y ∈ R g such that 0 ≠ x y ∈ P and y ∉ Grad ( P ) . We may assume that x , y are nonunit elements of R . Then, there is a nonunit w ∈ R g such that w x y ≠ 0 and w + u is a nonunit element of R g for some unit u ∈ R g . Since 0 ≠ w x y ∈ P and y ∉ Grad ( P ) and P is a g -weakly 1-absorbing primary ideal of R , we conclude that w x ∈ P . Since 0 ≠ ( w + u ) x y ∈ P and P is a g -weakly 1-absorbing primary ideal of R and y ∉ Grad ( P ) , we conclude that ( w + u ) x = w x + u x ∈ P . Since w x ∈ P and w x + u x ∈ P , we conclude that u x ∈ P . Since u is a unit, we have x ∈ P . Hence, P is a g -weakly primary ideal of R .□

Definition 2.7

Let R be a graded ring.

  1. For a , b ∈ h ( R ) , we say that a divides b (we write a ∣ b ) if b = a x for some x ∈ h ( R ) .

  2. R is said to be a graded chained ring if for every a , b ∈ h ( R ) , we have either a ∣ b or b ∣ a .

  3. R is said to be a graded weakly divided ring if for every graded weakly prime ideal P of R and for every a ∈ h ( R ) − P , we have a ∣ p for every p ∈ P .

Clearly, every graded chained ring is a graded weakly divided ring.

Proposition 2.8

Let R be a graded weakly divided ring and P be a proper graded ideal of R . If Grad ( { 0 } ) = { 0 } , then P is a graded weakly 1-absorbing primary ideal of R if and only if P is a graded weakly primary ideal of R .

Proof

Suppose that P is a graded weakly 1-absorbing primary ideal of R . Let x , y ∈ h ( R ) such that 0 ≠ x y ∈ P and y ∉ Grad ( P ) . We may assume that x , y are nonunit elements of R . Since Grad ( P ) is a graded weakly prime ideal of R by Proposition 2.3, we conclude that x ∈ Grad ( P ) . Since R is a graded weakly divided ring, we conclude that y ∣ x . Thus, x = y c for some c ∈ h ( R ) . Note that c is a nonunit element of R as y ∉ Grad ( P ) and x ∈ Grad ( P ) . Since 0 ≠ x y = y c y ∈ P and P is a graded weakly 1-absorbing primary ideal of R , and y ∉ Grad ( P ) , we conclude that y c = x ∈ P . Hence, P is a graded weakly primary ideal of R . The converse is clear.□

Corollary 2.9

Let R be a graded chained ring and P be a proper graded ideal of R. If Grad ( { 0 } ) = { 0 } , then P is a graded weakly 1-absorbing primary ideal of R if and only if P is a graded weakly primary ideal of R.

Let R be a G -graded ring and P be a graded ideal of R . Then, it has been proved in ([19], Theorem 2.17), ( P : a ) = { x ∈ R : x a ∈ P } is a graded ideal of R for every a ∈ h ( R ) . Similarly, one can prove that ( P : K ) is a graded ideal of R for every graded ideal K of R .

Proposition 2.10

Let R be a G -graded ring, g ∈ G and P be graded ideal of R with P g ≠ R g . Consider the following statements:

  1. P is a g -weakly 1-absorbing primary ideal of R .

  2. For every nonunit elements x , y ∈ R g with x y ∉ P , ( P : R g x y ) = ( 0 : R g x y ) or ( P : R g x y ) ⊆ Grad ( P ) .

  3. For every nonunit element x ∈ R g and every graded ideal K of R with K ⊈ Grad ( P ) , if ( P : R g x K ) ≠ R g , then ( P : R g x K ) = ( 0 : R g x K ) or ( P : R g x K ) ⊆ ( P : R g x ) .

  4. For every graded ideals K and J of R with K ⊈ Grad ( P ) , if ( P : R g K J ) ≠ R g , then ( P : R g K J ) = ( 0 : R g K J ) or ( P : R g K J ) ⊆ ( P : R g J ) .

Then, ( 1 ) ⇒ ( 2 ) ⇒ ( 3 ) ⇒ ( 4 ) .

Proof

( 1 ) ⇒ ( 2 ) : Suppose that x , y ∈ R g are nonunit elements such that x y ∉ P . Let c ∈ ( P : R g x y ) . Since x y ∉ P ; c is nonunit. If x y c = 0 , then c ∈ ( 0 : R g x y ) . Assume that 0 ≠ x y c ∈ P . Since P is a g -weakly 1-absorbing primary ideal of R , we have c ∈ Grad ( P ) . Hence, ( P : R g x y ) ⊆ ( 0 : R g x y ) ⋃ Grad ( P ) and so we obtain ( P : R g x y ) ⊆ ( 0 : R g x y ) or ( P : R g x y ) ⊆ Grad ( P ) . If ( P : R g x y ) ⊆ ( 0 : R g x y ) , then ( P : R g x y ) = ( 0 : R g x y ) since the reverse inclusion always holds. So, we obtain ( P : R g x y ) = ( 0 : R g x y ) or ( P : R g x y ) ⊆ Grad ( P ) .

( 2 ) ⇒ ( 3 ) : Let x ∈ R g be a nonunit element and K be a graded ideal of R with K ⊈ Grad ( P ) . If x K ⊆ P , then nothing to prove. Suppose that x K ⊈ P and let c ∈ ( P : R g x K ) . It is clear that c is nonunit. Then, x c K ⊆ P . Now K ⊆ ( P : R g x c ) . If x c ∈ P , then c ∈ ( P : R g x ) . Suppose that x c ∉ P . Hence, ( P : R g x c ) = ( 0 : R g x c ) or ( P : R g x c ) ⊆ Grad ( P ) by (2). Thus, K ⊆ ( 0 : R g x c ) or K ⊆ Grad ( P ) . Since K ⊈ Grad ( P ) , we conclude that K ⊆ ( 0 : R g x c ) ; that is c ∈ ( 0 : R g x K ) . Thus, ( P : R g x K ) ⊆ ( 0 : R g x K ) ⋃ ( P : R g x ) , and then we have ( P : R g x K ) = ( 0 : R g x K ) or ( P : R g x K ) ⊆ ( P : R g x ) .

( 3 ) ⇒ ( 4 ) : Let K and J be graded ideals of R with K ⊈ Grad ( P ) . Let c ∈ ( P : R g K J ) . Then, J ⊆ ( P : R g c K ) . Since ( P : R g K J ) ≠ R g , c is nonunit. So, J ⊆ ( 0 : R g c K ) or J ⊆ ( P : R g c ) by (3). If J ⊆ ( 0 : R g c K ) , then c ∈ ( P : R g K J ) . If J ⊆ ( P : R g c ) , then c ∈ ( P : R g J ) . So, ( P : R g K J ) ⊆ ( 0 : R g K J ) ⋃ ( P : R g J ) which implies that ( P : R g K J ) = ( 0 : R g K J ) or ( P : R g K J ) ⊆ ( P : R g J ) ; as required.□

Remark 2.11

Let R be a graded ring and P be a graded ideal of R . If P is a graded weakly 1-absorbing primary ideal of R that is not a graded 1- absorbing primary ideal of R , then it is clear that there exist nonunit elements x , y , z ∈ h ( R ) such that x y z = 0 with x y ∉ P and z ∉ Grad ( P ) . In fact, these x , y , z satisfy several properties as we see in the next results.

Let R be a G -graded ring, g ∈ G and P be a graded ideal of R such that P g ≠ R g . Then, P is said to be a g -1-absorbing primary ideal if whenever x , y , z ∈ R g such that x y z ∈ P , then either x y ∈ P or z ∈ Grad ( P ) .

Proposition 2.12

Let R be a G -graded ring, g ∈ G and P be a graded ideal of R . If P is a g -weakly 1-absorbing primary ideal of R that is not a g -1- absorbing primary ideal of R , then there exist nonunit elements x , y , z ∈ R g such that x y z = 0 with x y ∉ P and z ∉ Grad ( P ) and the following hold:

  1. x y P g = { 0 } .

  2. If x , y ∉ ( P : R g z ) , then y z P g = x z P g = x P g 2 = y P g 2 = z P g 2 = { 0 } .

  3. If x , y ∉ ( P : R g z ) , then P g 3 = { 0 } .

Proof

  1. Suppose that x y P g ≠ { 0 } . Then, x y a ≠ 0 for some nonunit a ∈ P g . So, 0 ≠ x y ( z + a ) ∈ P . Since x y ∉ P ; ( z + a ) is a nonunit element of R g . Since P is a g -weakly 1-absorbing primary ideal of R and x y ∉ P , we conclude that ( z + a ) ∈ Grad ( P ) . Since a ∈ P , we have z ∈ Grad ( P ) , which is a contradiction. So, x y P g = { 0 } .

  2. Suppose that y z P g ≠ { 0 } . Then, y z b ≠ 0 for some nonunit element b ∈ P g . Hence, 0 ≠ y z b = y ( x + b ) z ∈ P . Since y ∉ ( P : R g z ) , we conclude that x + b is a nonunit element of R g . Since P is a g -weakly 1-absorbing primary ideal of R and x y ∉ P and y b ∈ P , we conclude that y ( x + b ) ∉ P , and hence, z ∈ Grad ( P ) , which is a contradiction. Thus, y z P g = { 0 } . Suppose that x z P g ≠ { 0 } . Then, x z b ≠ 0 for some nonunit element b ∈ P g . Hence, 0 ≠ x z b = x ( y + b ) z ∈ P . Since x ∉ ( P : R g z ) , we conclude that y + b is a nonunit element of R g . Since P is a g -weakly 1-absorbing primary ideal of R and x y ∉ P and x b ∈ P , we conclude that x ( y + b ) ∉ P , and hence, z ∈ Grad ( P ) , which is a contradiction. Thus, x z P g = { 0 } . Suppose that x P g 2 ≠ { 0 } . Then, x a b ≠ 0 for some a ; b ∈ P g . Since x y P g = { 0 } by (1) and x z P g = { 0 } by (2), 0 ≠ x a b = x ( y + a ) ( z + b ) ∈ P . Since x y ∉ P , we conclude that z + b is a nonunit element of R g . Since x ∉ ( P : R g z ) , we conclude that y + a is a nonunit element of R g . Since P is a g -weakly 1-absorbing primary ideal of R , we have x ( y + a ) ∈ P or ( z + b ) ∈ Grad ( P ) . Since a ; b ∈ P , we conclude that x y ∈ P or z ∈ Grad ( P ) , which is a contradiction. Thus, x P g 2 = { 0 } . Suppose that y P g 2 ≠ { 0 } . Then, y a b ≠ 0 for some a ; b ∈ P g . Since x y P g = { 0 } by (1) and y z P g = { 0 } by (2), 0 ≠ y a b = y ( x + a ) ( z + b ) ∈ P . Since x y ∉ P , we conclude that z + b is a nonunit element of R g . Since y ∉ ( P : R g z ) , we conclude that x + a is a nonunit element of R g . Since P is a g -weakly 1-absorbing primary ideal of R , we have y ( x + a ) ∈ P or ( z + b ) ∈ Grad ( P ) . Since a ; b ∈ P , we conclude that x y ∈ P or z ∈ Grad ( P ) , which is a contradiction. Thus, y P g 2 = { 0 } . Suppose that z P g 2 ≠ { 0 } . Then, z a b ≠ 0 for some a ; b ∈ P g . Since x z P g = y z P g = { 0 } by (2), 0 ≠ z a b = ( x + a ) ( y + b ) z ∈ P . Since x ; y ∉ ( P : R g z ) , we conclude that x + a and y + b are nonunit elements of R g . Since P is a g -weakly 1-absorbing primary ideal of R , we have ( x + a ) ( y + b ) ∈ P or z ∈ Grad ( P ) . Since a ; b ∈ P , we conclude that x y ∈ P or z ∈ Grad ( P ) , which is a contradiction. Thus, z P g 2 = { 0 } .

  3. Suppose that P g 3 ≠ { 0 } . Then, a b c ≠ 0 for some a , b , c ∈ P g . Then, 0 ≠ a b c = ( x + a ) ( y + b ) ( z + c ) ∈ P by (1) and (2). Since x y ∉ P , we conclude z + c is a nonunit element of R g . Since x ; y ∉ ( P : R g z ) , we conclude that x + a and y + b are nonunit elements of R g . Since P is a g -weakly 1-absorbing primary ideal of R , we have ( x + a ) ( y + b ) ∈ P or z + c ∈ Grad ( P ) . Since a ; b ; c ∈ P , we conclude that x y ∈ P or z ∈ Grad ( P ) , which is a contradiction. Thus, P g 3 = { 0 } .□

Corollary 2.13

Let R be a G-graded ring, g ∈ G and P be a graded ideal of R. If P is a g-weakly 1-absorbing primary ideal of R that is not a g-1-absorbing primary ideal of R, then there exist nonunit elements x , y , z ∈ R g such that x y z = 0 with x y ∉ P and z ∉ Grad ( P ) . If x , y ∉ ( P : R g z ) and Grad ( { 0 } ) = { 0 } , then P g = { 0 } .

Proof

By Proposition 2.12 (3), we have P g 3 = { 0 } , and since Grad ( { 0 } ) = { 0 } , we conclude that P g = { 0 } .□

Definition 2.14

Let R be a graded ring. Then, x ∈ h ( R ) is said to be a homogeneous reducible element of R if x = y z for some nonunit elements y , z ∈ h ( R ) . Otherwise, x is called a homogeneous irreducible element of R .

Proposition 2.15

Let P be a graded weakly 1-absorbing primary ideal of a graded ring R . If P is not a graded weakly primary ideal of R , then there exist a homogeneous irreducible element a ∈ R and a nonunit element b ∈ h ( R ) such that a b ∈ P , but neither a ∈ P nor b ∈ Grad ( P ) . Moreover, if x y ∈ P for some nonunit elements x , y ∈ h ( R ) such that neither x ∈ P nor y ∈ Grad ( P ) , then x is a homogeneous irreducible element of R .

Proof

Since P is not a graded weakly primary ideal of R , we conclude that there exist nonunit elements a ; b ∈ h ( R ) such that 0 ≠ a b ∈ P with a ∉ P and b ∉ Grad ( P ) . Suppose that a is a homogeneous reducible element of R . Then, a = c d for some nonunit elements c ; d ∈ h ( R ) . Since 0 ≠ a b = c d b ∈ P and P is a graded weakly 1-absorbing primary ideal of R and b ∉ Grad ( P ) , we conclude that c d = a ∈ P , which is a contradiction. Hence, a is a homogeneous irreducible element of R .□

Proposition 2.16

Let R be a graded ring and P 1 , P 2 , … , P n be graded weakly 1-absorbing primary ideals of R . If Grad ( P i ) = Grad ( P j ) = Q for every i , j , then P = ⋂ i = 1 n P i is a graded weakly-1-absorbing primary ideal of R .

Proof

Suppose that x , y , z ∈ h ( R ) are nonunit elements such that 0 ≠ x y z ∈ P . Suppose that x y ∉ P . Then, x y ∉ P k for some 1 ≤ k ≤ n . Since P k is a graded weakly 1-absorbing primary ideal of R and 0 ≠ x y z ∈ P k and x y ∉ P k , we have that z ∈ Grad ( P k ) = Q = Grad ( P ) . Hence, P is a graded weakly 1-absorbing primary ideal of R .□

Proposition 2.17

Let P be a graded weakly 1-absorbing primary ideal of a graded ring R . Then, ( P : z ) is a graded weakly primary ideal of R for every nonunit z ∈ h ( R ) − P .

Proof

Let z ∈ h ( R ) − P be a nonunit element. Then, ( P : z ) is a graded ideal of R . Let x , y ∈ h ( R ) such that 0 ≠ x y ∈ ( P : z ) , assume that x ∉ ( P : z ) . Hence, y is a nonunit element of R . If x is an unit element of R , then y ∈ ( P : z ) ⊆ Grad ( ( P : z ) ) and we are done. Assume that x is a nonunit element of R . Since 0 ≠ x y z = x z y ∈ P and x z ∉ P and P is a graded weakly-1-absorbing primary ideal of R , we conclude that y ∈ Grad ( P ) ⊆ Grad ( ( P : z ) ) . Thus, ( P : z ) is a graded weakly primary ideal of R .□

A G -graded ring R is said to be a cross product if R g contains a unit element for all g ∈ G [16].

Proposition 2.18

Let R = S × T , where S and T are G -graded commutative rings with a nonzero unity 1 that are not graded fields. Suppose that R and S are cross products. Assume that P is a nonzero proper graded ideal of R . Then, the following assertions are equivalent:

  1. P is a graded weakly 1-absorbing primary ideal of R .

  2. P = I × T for some graded primary ideal I of S or P = S × J for some graded primary ideal J of T.

  3. P is a graded primary ideal of R .

  4. P is a graded 1-absorbing primary ideal of R .

Proof

( 1 ) ⇒ ( 2 ) : Now, P is of the form I × J for some graded ideals I and J of S and T , respectively. Assume that both I and J are proper. Since P is a nonzero ideal of R , we conclude that I or J is nonzero. We may assume that I is nonzero. Let 0 ≠ z ∈ I . Then, since I is graded, 0 ≠ z g ∈ I for some g ∈ G . Since T is a cross product, choose a unit homogeneous element t g ∈ T g . Then, 0 ≠ ( 1 , 0 ) ( 1 , 0 ) ( z g , t g ) = ( z g , 0 ) ∈ P . It implies that ( 1 , 0 ) ( 1 , 0 ) ∈ P or ( z g , t g ) ∈ Grad ( P ) = Grad ( I ) × Grad ( J ) , that is I = S or J = T , a contradiction. Thus, I = S or J = T . Without loss of generality, assume that P = I × T for some proper graded ideal I of S . We show that I is a graded primary ideal of S . Let x y ∈ I for some x , y ∈ h ( S ) , where deg ( x ) = g and deg ( y ) = h . We can assume that x and y are nonunit elements of S . Since T is not a graded field, there exists a nonunit nonzero element a ∈ h ( T ) , where deg ( a ) = j . Since S and T are cross products, there exist homogeneous unit elements s ∈ S and t , t ′ ∈ T , where deg ( s ) = j , deg ( t ) = g and deg ( t ′ ) = h . Then, 0 ≠ ( x , t ) ( s , a ) ( y , t ′ ) = ( s x y , t a t ′ ) ∈ I × T , which implies that either ( x , t ) ( s , a ) ∈ I × T or ( y , t ′ ) ∈ Grad ( I × T ) = Grad ( I ) × Grad ( T ) , that is, x ∈ I or y ∈ Grad ( I ) .

( 2 ) ⇒ ( 3 ) : It is well-known.

( 3 ) ⇒ ( 4 ) and ( 4 ) ⇒ ( 1 ) are clear.□

Proposition 2.19

Let R = R 1 × R 2 × ⋯ × R n , where R 1 , R 2 , … , R n are G -graded commutative rings with a nonzero unity 1. Suppose that R i is a cross product, for every i = 1 , 2 , … , n . Then, every proper graded ideal of R is a graded weakly 1-absorbing primary ideal of R if and only if n = 2 and R 1 , R 2 are graded fields.

Proof

Suppose that every proper graded ideal of R is a graded weakly 1-absorbing primary ideal of R . Without loss of generality, we may assume that n = 3 . Then, P = R 1 × { 0 } × { 0 } is a graded weakly 1-absorbing primary ideal of R . Choose a nonzero x ∈ h ( R 1 ) , where deg ( x ) = g . Since R 2 is a cross product, there exists a unit homogeneous element t ∈ h ( R 2 ) , where deg ( t ) = g . Then, 0 ≠ ( 1 , 0 , 1 ) ( 1 , 0 , 1 ) ( x , t , 0 ) = ( x , 0 , 0 ) ∈ P , but ( 1 , 0 , 1 ) ( 1 , 0 , 1 ) ∈ P nor ( x , t , 0 ) ∈ Grad ( P ) , a contradiction. Thus, n = 2 . Assume that R 1 is not a graded field. Then, there exists a nonzero proper graded ideal I of R 1 . Hence, P = I × { 0 } is a graded weakly 1-absorbing primary ideal of R . For a nonzero x ∈ I , 0 ≠ x g ∈ I for some g ∈ G . Since R 2 is a cross product, there exists a unit homogeneous element u ∈ h ( R 2 ) , where deg ( u ) = g . Then, we have 0 ≠ ( 1 , 0 ) ( 1 , 0 ) ( x g , u ) = ( x g , 0 ) ∈ P , but ( 1 , 0 ) ( 1 , 0 ) ∈ P nor ( x g , u ) ∈ Grad ( P ) , a contradiction. Similarly, R 2 is a graded field. Conversely, R has exactly three proper graded ideals; { ( 0 , 0 ) } , { 0 } × R 2 and R 1 × { 0 } , and all of them are graded weakly 1-absorbing primary ideals of R .□

First strongly graded rings have been introduced and studied in [20]; a G -graded ring R is said to be first strong if 1 ∈ R g R g − 1 for all g ∈ supp ( R , G ) , where supp ( R , G ) = { g ∈ G : R g ≠ { 0 } } . In fact, it has been proved that R is first strongly G -graded if and only if supp ( R , G ) is a subgroup of G and R g R h = R g h for all g , h ∈ supp ( R , G ) . We introduce the following:

Lemma 2.20

Every G -graded field is first strongly graded.

Proof

Let R be a G -graded field. Suppose that g ∈ supp ( R , G ) . Then, R g ≠ { 0 } , and then there exists 0 ≠ x ∈ R g . Since R is a graded field, we conclude that there exists y ∈ h ( R ) such that x y = 1 . Since y ∈ h ( R ) , y ∈ R h for some h ∈ G , and then 1 = x y ∈ R g R h ⊆ R g h . So, 0 ≠ 1 ∈ R g h ⋂ R e , which implies that g h = e , that is h = g − 1 . Hence, 1 = x y ∈ R g R g − 1 , and thus, R is first strongly graded.□

Proposition 2.21

Let R = R 1 × R 2 , where R 1 and R 2 are G -graded commutative rings with a nonzero unity 1. Suppose that R i is a cross product, for every i = 1 , 2 . If every proper graded ideal of R is a graded weakly 1-absorbing primary ideal of R , then R 1 and R 2 are first strongly graded.

Proof

Apply Proposition 2.19 and Lemma 2.20.□

Let R and S be two G -graded rings. A ring homomorphism f : R → S is said to be graded homomorphism if f ( R g ) ⊆ S g for all g ∈ G .

Proposition 2.22

Let R and S be G-graded commutative rings and f : R → S is a graded homomorphism with f ( 1 R ) = 1 S . Then, the following hold:

  1. Suppose that f ( x ) is a nonunit element of S for every nonunit element x ∈ R and I is a graded weakly 1-absorbing primary ideal of S. Then, f − 1 ( I ) is a graded weakly 1-absorbing primary ideal of R.

  2. If f is surjective and P is a graded weakly 1-absorbing primary ideal of R such that Ker ( f ) ⊆ P , then f ( P ) is a graded weakly 1-absorbing primary ideal of S.

Proof

  1. Clearly, f − 1 ( I ) is a graded ideal of R . Let x , y , z ∈ h ( R ) be nonunit elements such that 0 ≠ x y z ∈ f − 1 ( I ) . Then, f ( x ) , f ( y ) , f ( z ) ∈ h ( S ) are nonunit elements such that 0 ≠ f ( x ) f ( y ) f ( z ) = f ( x y z ) ∈ I . Since I is a graded weakly 1-absorbing primary ideal of S , we have that f ( x y ) = f ( x ) f ( y ) ∈ I or f ( z ) ∈ Grad ( I ) , which implies that x y ∈ f − 1 ( I ) or z ∈ f − 1 ( Grad ( I ) ) = Grad ( f − 1 ( I ) ) . Thus, f − 1 ( I ) is a graded weakly 1-absorbing primary ideal of R .

  2. Clearly, f ( P ) is a graded ideal of S . Let a , b , c ∈ h ( S ) be nonunit elements such that 0 ≠ a b c ∈ f ( P ) . Then, since f is surjective, there exist nonunit elements x , y , z ∈ h ( R ) such that f ( x ) = a , f ( y ) = b and f ( z ) = c . Now, f ( x y z ) = f ( x ) f ( y ) f ( z ) = a b c ∈ f ( P ) . Since Ker ( f ) ⊆ P , we have that 0 ≠ x y z ∈ P . Since P is a graded weakly 1-absorbing primary ideal of R , we have that x y ∈ P or z ∈ Grad ( P ) , which implies that a b = f ( x ) f ( y ) = f ( x y ) ∈ f ( P ) or c = f ( z ) ∈ f ( Grad ( P ) ) = Grad ( f ( P ) ) as f is surjective and Ker ( f ) ⊆ P . Hence, f ( P ) is a graded weakly 1-absorbing primary ideal of S .□

Corollary 2.23

Let P be a proper graded ideal of a graded ring R. Then, the following hold:

  1. If K is a proper graded ideal of R with K ⊆ P and P is a graded weakly 1-absorbing primary ideal of R, then P ∕ K is a graded weakly 1-absorbing primary ideal of R ∕ K .

  2. Let K be a proper graded ideal of R with K ⊆ P such that U ( R ∕ K ) = { x + K : x ∈ U ( R ) } . If K is a graded 1-absorbing primary ideal of R and P ∕ K is a graded weakly 1-absorbing primary ideal of R ∕ K , then P is a graded 1-absorbing primary ideal of R.

  3. If { 0 } is a graded 1-absorbing primary ideal of R and P is a graded weakly 1-absorbing primary ideal of R, then P is a graded 1-absorbing primary ideal of R.

  4. Let K be a proper graded ideal of R with K ⊆ P such that U ( R ∕ K ) = { x + K : x ∈ U ( R ) } . If K is a graded weakly 1-absorbing primary ideal of R and P ∕ K is a graded weakly 1-absorbing primary ideal of R ∕ K , then P is a graded weakly 1-absorbing primary ideal of R .

Proof

  1. Let f : R → R ∕ K such that f ( x ) = x + K . Then, f is surjective graded homomorphism and f ( 1 R ) = 1 R ∕ K . Suppose that P is a graded weakly 1-absorbing primary ideal of R . Since f is surjective and Ker ( f ) = K ⊆ P , by Proposition 2.22(2), we have that f ( P ) = P ∕ K is a graded weakly 1-absorbing primary ideal of R ∕ K .

  2. Suppose that x y z ∈ P for some nonunit elements x , y , z ∈ h ( R ) . If x y z ∈ K , then x y ∈ K ⊆ P or z ∈ Grad ( K ) ⊆ Grad ( P ) as K is a graded 1-absorbing primary ideal of R . Now, assume that x y z ∉ K . Then, K ≠ ( x + K ) ( y + K ) ( z + K ) ∈ P ∕ K , where x + K , y + K , z + K are nonunit elements of h ( R ∕ K ) by assumption. Thus, ( x + K ) ( y + K ) ∈ P ∕ K or ( z + K ) ∈ Grad ( P ∕ K ) . Hence, x y ∈ P or z ∈ Grad ( P ) .

  3. Apply (2).

  4. Suppose that 0 ≠ x y z ∈ P for some nonunit elements x , y , z ∈ h ( R ) . If x y z ∈ K , then x y ∈ K ⊆ P or z ∈ Grad ( K ) ⊆ Grad ( P ) as K is a graded weakly 1-absorbing primary ideal of R . Now, assume that x y z ∉ K . Then, K ≠ ( x + K ) ( y + K ) ( z + K ) ∈ P ∕ K , where x + K , y + K , z + K are nonunit elements of h ( R ∕ K ) by assumption. Thus, ( x + K ) ( y + K ) ∈ P ∕ K or ( z + K ) ∈ Grad ( P ∕ K ) . Hence, x y ∈ P or z ∈ Grad ( P ) .□

3 Conclusion

In this article, we follow [6] to introduce and study the concept of graded weakly 1-absorbing primary ideal of a graded commutative ring R . It is recognizable that a graded 1-absorbing primary ideal of R is a graded weakly 1-absorbing primary ideal of R . However, since { 0 } is always graded weakly 1-absorbing primary, a graded weakly 1-absorbing primary ideal of R needs not be a graded 1-absorbing primary ideal of R (see Example 2.2). Among several results, we show that if P is a graded weakly 1-absorbing primary ideal of R and Grad ( { 0 } ) = { 0 } , then Grad ( P ) is a graded weakly prime ideal of R (Proposition 2.3). We prove that if P is a graded weakly 1-absorbing primary ideal of R , then ( P : z ) is a graded weakly primary ideal of R for every nonunit z ∈ h ( R ) − P (Proposition 2.17). We show that if R = R 1 × R 2 × ⋯ × R n , where R 1 , R 2 , … , R n are graded rings, then every proper graded ideal of R is a graded weakly 1-absorbing primary ideal of R if and only if n = 2 and R 1 , R 2 are graded fields (Proposition 2.19). In Proposition 2.22, we study graded weakly 1-absorbing primary ideals over graded homomorphisms. Concerning the weakness of the developed work, we should recover it by finding several applications in several areas, and that will be studied as a proposal for a future work.

Acknowledgements

We would like to express our sincere gratitude to the referees for their valuable time and effort in reviewing our research article. Their insightful comments and suggestions have greatly contributed to the improvement of our work. We appreciate their expertise and dedication to the scientific community. Thank you.

  1. Conflict of interest: The authors declare no conflict of interest.

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Received: 2021-05-14
Revised: 2023-01-22
Accepted: 2023-02-16
Published Online: 2023-07-19

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2022-0214
Keywords for this article
graded prime ideal; graded weakly prime ideal; graded primary ideal; graded weakly primary ideal; graded 2-absorbing primary ideal; graded weakly 2-absorbing primary ideal; graded 2-absorbing ideal; graded weakly 2-absorbing ideal; graded n-absorbing ideal
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  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
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  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
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  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
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  88. Petri net analysis of a queueing inventory system with orbital search by the server
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  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
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