Graded I - second submodules

: Let G be a group with identity e , R be a G - graded commutative ring with a nonzero unity 1, I be a graded ideal of R , and M be a G - graded R - module. In this article, we introduce the concept of graded I - second submodules of M as a generalization of graded second submodules of M and achieve some relevant outcomes.


Introduction
A proper graded ideal P of R is said to be graded prime if whenever ∈ ( ) x y h R , such that ∈ xy P, then either ∈ x P or ∈ y P. Graded prime ideals have been admirably introduced and studied in [1]. Graded prime submodules have been introduced by Atani in [2]. A proper graded R-submodule N of M is said to be graded prime if whenever ∈ ( ) r h R and ∈ ( ) m h M such that ∈ rm N , then either ∈ m N or ∈ ( ) r N M : R . Graded prime submodules have been widely studied by several authors, for more details one can look in [3][4][5][6]. Atani introduced in [7] 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 ≠ ∈ xy P 0 , then ∈ x P or ∈ y P. Also, Atani extended the concept of graded weakly prime ideals into graded weakly prime submodules in [8]. A proper graded submodule N of M is called graded weakly prime if for ∈ ( ) r h R and ∈ ( ) m h M with ≠ ∈ rm N 0 , either ∈ m N or ∈ ( ) r N M : R . Let M and S be two G-graded R-modules. An R-homomorphism → f M S : is said to be graded R-homomorphism if ( ) ⊆ f M S g g for all ∈ g G (see [9]). Graded second submodules have been introduced by Ansari-Toroghy and Farshadifar in [10]. A nonzero graded R-submodule N of M is said to be graded second if for each ∈ ( ) a h R , the graded R-homomorphism → f N N : defined by ( ) = f x ax is either surjective or zero. In this case, ( ) N Ann R is a graded prime ideal of R. Graded second submodules have been wonderfully studied by Çeken and Alkan in [11]. On the other hand, graded secondary modules have been introduced by Atani and Farzalipour in [12]. A nonzero graded R-module M is said to be graded secondary if for each ∈ ( ) a h R , the graded R-homomorphism → f M M : defined by ( ) = f x ax is either surjective or nilpotent. The main purpose of this article is to follow [13] in order to introduce and study the concept of graded I-second submodules of a graded R-module M as a generalization of graded second submodules of M and achieve some relevant outcomes. Among several results, we show that a graded second submodule is a graded I-second submodule for every graded ideal I of R, but we prove that the converse is not true in general (Examples 2.5, 2.6, and 2.7). We follow [14] to introduce the concept of graded I-prime ideals of a graded ring R, we show that a graded prime ideal is a graded I-prime ideal for every graded ideal I of R, but we prove that the converse is not true in general (Example 2.16). We prove that if N is a graded I-second R-submodule of M such that (( )) ⊆ ( ) is a graded I-prime ideal of R (Proposition 2.21). We show that if M is a graded comultiplication R-module and N is a graded R-submodule of M such that ( ) N Ann R is an I-prime ideal of R, then N is a graded I-second R-submodule of M (Proposition 2.23). We prove that if M is primary, then every proper graded { } 0 -second R-submodule of M is a graded primary R-submodule of M (Proposition 2.27). In Proposition 2.28, we study graded I-second submodules under graded homomorphism. Finally, in Proposition 2.29, we study the relation between graded I-second submodules of M and I e -second submodules of M e when | | = G 2.

Preliminaries
Throughout this article, G will be a group with identity e and R will be a commutative ring with a nonzero unity 1. R is said to be G-graded if , where R g is an additive subgroup of R for all ∈ g G. The elements of R g are called homogeneous of degree g. Consider Moreover, it has been proved in [9] that R e is a subring of R and ∈ R 1 e . Let I be an ideal of a graded ring R. Then I is said to be a graded ideal if Let R be a G-graded ring and I be a graded ideal of R. Then R/I is , where M g is an additive subgroup of M for all ∈ g G. The elements of M g are called homogeneous of degree g. Also, we consider Similarly, if M is a graded R-module, N a graded R-submodule of M, and ∈ ( ) m h M , then ( ) N m : R is a graded ideal of R. Also, it has been proved in [16] In [16], the concept of graded completely irreducible submodules has been extended into G-graded case, for any group G. It has been proved that every graded R-submodule of M is an intersection of graded completely irreducible R-submodules of M. In many instances, we use the following basic fact without further discussion.
Remark 1.2. Let N and L be two graded R-submodules of M. To prove that ⊆ N L, it is enough to prove that if K is a graded completely irreducible R-submodule of M such that ⊆ L K , then ⊆ N K .

Graded I-second submodules
In this section, we introduce and study the concept of graded I-second submodules.
be the set of all graded completely irreducible R-submodules of M. Assume that P is a graded prime ideal of R and N is a graded R-submodule of M. Then we define ( ) = ⋂ ∈ ( ) The following lemma gives some characterizations for graded second R-submodules.
Lemma 2.1. Let N be a graded R-submodule of a graded R-module M. Then the following are equivalent.
Proposition 2.3. Let M be a graded R-module, I be a graded ideal of R, and N be a nonzero graded R-submodule of M. Then the following statements are equivalent: : : A n n : : : : : : , which is a contradiction. Thus, ∉ ( ( )) r rN N I : : : : A n n : : . The other inclusion is clear. Clearly, every graded second submodule is a graded I-second submodule for every graded ideal I of R. However, the following examples prove that the converse is not true in general. : : Definition 2.12. Let M be a G-graded R-module, I be a graded ideal of R, N be a nonzero graded R-submodule of M, and ∈ g G. Then N is said to be a g-I-second R-submodule of M if for each ∈ r R g , and a graded R-submodule K of M, ∈ ( ) − ( ( )) r K N K N I : : : Definition 2.13. Let M be a G-graded R-module and ∈ g G. A nonzero graded R-submodule N of M is said to be a g-second R-submodule of M if K is a graded R-submodule of M and ∈ r R g such that ⊆ rN K , then either = { } rN 0 or ⊆ N K .
Proposition 2.14. Let M be a G-graded R-module and ∈ g G. If N is a g-I-second R-submodule of M which is not graded g-second, then . Then + ∈ ( ) − ( ( )) t r K N K N I : : : : : In the following definition, we follow [14] to introduce the concept of graded I-prime ideals of a graded ring R.
Definition 2.15. Let R be a graded ring and I be a graded ideal of R. Then a proper graded ideal P of R is said to be graded I-prime if for ∈ ( ) x y h R , such that ∈ − xy P IP, then either ∈ x P or ∈ y P.
Clearly, every graded prime ideal is a graded I-prime ideal for every graded ideal I of R. However, the following example shows that the converse is not true in general.
. If we take = = 〈 〉 P I 4 , then P is a graded I-prime ideal of R which is neither graded prime nor graded weakly prime.
Lemma 2.17. Let R be a G-graded ring, I be an ideal of R, and J be a graded ideal of R such that ⊆ J I . Then I is a graded ideal of R if and only if I/J is a graded ideal of R/J.
Proof. Suppose that I is a graded ideal of R. Clearly, I/J is an ideal of R/J. Let + ∈ / x J I J . Then ∈ x I and since I is graded, = ∑ ∈ x x g G g , where ∈ x I g for all ∈ g G and then ( + ) = + ∈ / x J x J I J g g for all ∈ g G. Hence, I/J is a graded ideal of R/J. Conversely, let ∈ x I. Then = ∑ ∈ x x g G g , where ∈ x R g g for all ∈ g G and then Since I/J is graded, . Then ∈ ( ) x y h R , such that ∈ − xy P IP, and then ∈ x P or ∈ y P. So, + ∈ / x IP P IP or + ∈ / y IP P IP. Hence, P/IP is a graded weakly prime ideal of R/IP. Conversely, let ∈ ( ) x y h R , such that ∈ − xy P IP. Then + x IP, + ∈ ( / ) y IP h R IP such that + ≠ IP 0 ( + )( + ) ∈ / x IP y IP P IP, and then + ∈ / x IP P IP or + ∈ / y IP P IP. So, ∈ x P or ∈ y P. Hence, P is a graded I-prime ideal of R. □ Proposition 2.19. Let I and J be two graded ideals of R such that ⊆ I J . Then every graded I-prime ideal of R is graded J-prime.
Proof. Let P be a graded I-prime ideal of R. Then the result follows from the fact that − ⊆ − P JP P IP. □ The following example shows that if I and J are two graded ideals of R such that ⊆ I J and P is a graded J-prime ideal of R, then P does not need to be graded I-prime. Proof. By [16], . This implies that ∉ (( ) ( )) x y N I 0 : : : . The concept of graded comultiplication modules has been studied by several authors, for example, see [19,20]. Proof. Let ∈ ( ) − ( ( )) r K N K N I : : : for some ∈ ( ) r h R and a graded R-submodule K of M. As M is a graded comultiplication R-module, there exists a graded ideal J of R such that = ( )  Graded primary ideals have been introduced and studied in [21]. A proper graded ideal P of R is said to be graded primary if for ∈ ( ) x y h R , such that ∈ xy P, then either ∈ x P or ∈ ( ) y P Grad , where ( ) P Grad is the graded radical of 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 that satisfies ∈ r P is said to be graded R-homomorphism if ( ) ⊆ f M S g g for all ∈ g G (see [9]). , which is a desired contradiction. □