Fatemeh Soheilnia - Academia.edu (original) (raw)

Papers by Fatemeh Soheilnia

International Electronic Journal of Algebra, Jan 5, 2021

Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential ... more Let R be a commutative ring with nonzero identity and let M be a unitary R-module. The essential graph of M , denoted by EG(M) is a simple undirected graph whose vertex set is Z(M)\Ann R (M) and two distinct vertices x and y are adjacent if and only if Ann M (xy) is an essential submodule of M. Let r(Ann R (M)) = Ann R (M). It is shown that EG(M) is a connected graph with diam(EG(M)) ≤ 2. Whenever M is Noetherian, it is shown that EG(M) is a complete graph if and only if either Z(M) = r(Ann R (M)) or EG(M) = K 2 and diam(EG(M)) = 2 if and only if there are x, y ∈ Z(M)\Ann R (M) and p ∈ Ass R (M) such that xy ∈ p. Moreover, it is proved that gr(EG(M)) ∈ {3, ∞}. Furthermore, for a Noetherian module M with r(Ann R (M)) = Ann R (M) it is proved that |Ass R (M)| = 2 if and only if EG(M) is a complete bipartite graph that is not a star.

Second Seminar on Algebra and its Applications

In this talk, all rings are commutative with nonzero identity and all modules are considered to b... more In this talk, all rings are commutative with nonzero identity and all modules are considered to be unitary. Prime submodules have an important role in the module theory over commutative rings. We recall that a proper submodule N of R-module M is called a prime submodule if whenever a∈ R and m∈ M with am∈ N, then either m∈ N or a∈(N: RM).

International Electronic Journal of Algebra

Let RRR be a commutative ring with nonzero identity and let MMM be a unitary RRR-module. The esse... more Let RRR be a commutative ring with nonzero identity and let MMM be a unitary RRR-module. The essential graph of MMM, denoted by EG(M)EG(M)EG(M) is a simple undirected graph whose vertex set is Z(M)setminusrmAnnR(M)Z(M)\setminus {\rm Ann}_R(M)Z(M)setminusrmAnnR(M) and two distinct vertices xxx and yyy are adjacent if and only if rmAnnM(xy){\rm Ann}_{M}(xy)rmAnnM(xy) is an essential submodule of MMM. Let r(rmAnnR(M))not=rmAnnR(M)r({\rm Ann}_R(M))\not={\rm Ann}_R(M)r(rmAnnR(M))not=rmAnnR(M). It is shown that EG(M)EG(M)EG(M) is a connected graph with rmdiam(EG(M))leq2{\rm diam}(EG(M))\leq 2rmdiam(EG(M))leq2. Whenever MMM is Noetherian, it is shown that EG(M)EG(M)EG(M) is a complete graph if and only if either Z(M)=r(rmAnnR(M))Z(M)=r({\rm Ann}_R(M))Z(M)=r(rmAnnR(M)) or EG(M)=K2EG(M)=K_{2}EG(M)=K2 and rmdiam(EG(M))=2{\rm diam}(EG(M))= 2rmdiam(EG(M))=2 if and only if there are x,yinZ(M)setminusrmAnnR(M)x, y\in Z(M)\setminus {\rm Ann}_R(M)x,yinZ(M)setminusrmAnnR(M) and frakpinrmAssR(M)\frak p\in{\rm Ass}_R(M)frakpinrmAssR(M) such that xynotinfrakpxy\not \in \frak pxynotinfrakp. Moreover, it is proved that rmgr(EG(M))in3,infty{\rm gr}(EG(M))\in \{3, \infty\}rmgr(EG(M))in3,infty. Furthermore, for a Noetherian module MMM with r(rmAnnR(M))=rmAnnR(M)r({\rm Ann}_R(M))={\rm Ann}_R(M)r(rmAnnR(M))=rmAnnR(M) it is proved that ∣rmAssR(M)∣=2|{\rm Ass}_R(M)|=2∣rmAssR(M)∣=2 if and only if EG(M)EG(M)EG(M) is a complete bipartite graph that is not...

All rings are commutative with identity, and all modules are unital. The purpose of this article ... more All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate n-absorbing submodules. For this reason we introduce the concept of n-absorbing submodules generalizing n-absorbing ideals of rings. Let M be an R-module. A proper submodule N of M is called an n-absorbing submodule if whenever a 1 • • • a n m ∈ N for a 1 ,. .. , a n ∈ R and m ∈ M , then either a 1 • • • a n ∈ (N : R M) or there are n − 1 of a i 's whose product with m is in N. We study the basic properties of n-absorbing submodules and then we study n-absorbing submodules of some classes of modules (e.g. Dedekind modules, Prüfer modules, etc.) over commutative rings.

Thai Journal of Mathematics, 2012

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 6= abm ∈ N), then ab ∈ (N :R M) or am ∈ N or bm ∈ N . It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P1 ∩ P2, P1P2 ⊆ N and (M − radN) ⊆ N where P1, P2 are the only distinct minimal prime submodules of N .

Let G be an arbitrary group with identity e and let R be a G-graded ring. In this paper, we defin... more Let G be an arbitrary group with identity e and let R be a G-graded ring. In this paper, we define the concept of graded 2-absorbing and graded weakly 2-absorbing primary ideals of commutative G-graded rings with non-zero identity. A number of results and basic properties of graded 2-absorbing primary and graded weakly 2-absorbing primary ideals are given.

Let R be a commutative ring with identity 1 6 0. Various generalizations of prime ideals have bee... more Let R be a commutative ring with identity 1 6 0. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R isweakly prime if a;b2 R with 06 ab2 I, then either a2 I or b2 I. Also a proper ideal I of R is said to be 2-absorbing if whenever a;b;c 2 R and abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. In this paper, we introduce the concept of a weakly 2-absorbing ideal. A proper ideal I of R is called a weakly 2-absorbing ideal of R if whenever a;b;c2 R and 0 6 abc 2 I, then either ab 2 I or ac 2 I or bc 2 I. For example, every proper ideal of a quasi-local ring (R;M) with M 3 =f0g is a weakly 2-absorbing ideal of R. We show that a weakly 2-absorbing ideal I of R with I 3 6 0 is a 2-absorbing ideal of R. We show that every proper ideal of a commutative ring R is a weakly 2-absorbing ideal if and only if either R is a quasi-local ring with maximal ideal M such that M 3 =f0g or R is ring- isomorphic to R1 F where R1 is a quasi-local ring with maximal...

All rings are commutative with identity, and all modules are unital. The pur- pose of this articl... more All rings are commutative with identity, and all modules are unital. The pur- pose of this article is to investigate n-absorbing submodules. For this reason we introduce the concept of n-absorbing submodules generalizing n-absorbing ideals of rings. Let M be an R-module. A proper submodule N of M is called an n-absorbing submodule if when- ever a1 ¢¢¢anm 2 N for a1;:::;an 2 R and m 2 M, then either a1 ¢¢¢an 2 (N :R M) or there are n i 1 of ai's whose product with m is in N. We study the basic properties of n-absorbing submodules and then we study n-absorbing submodules of some classes of modules (e.g. Dedekind modules, Prufer modules, etc.) over commutative rings. AMS subject classiflcations: 13A15, 13F05

All rings are commutative with 1 6= 0. The purpose of this paper is to investigate the concept of... more All rings are commutative with 1 6= 0. The purpose of this paper is to investigate the concept of weakly n-absorbing ideals generalizing weakly 2-absorbing ideals. We prove that over a u-ring R the Anderson-Badawi’s conjectures about n-absorbing ideals and the Badawi-Yousefian’s question about weakly 2-absorbing ideals hold.

Kyungpook mathematical journal, 2016

Mathematical Communications

Mathematical Communications, Dec 5, 2012

Sažetak All rings are commutative with identity, and all modules are unital. The purpose of this ... more Sažetak All rings are commutative with identity, and all modules are unital. The purpose of this article is to investigate n-absorbing submodules. For this reason we introduce the concept of n-absorbing submodules generalizing n-absorbing ideals of rings. Let M be an R-module. A proper submodule N of M is called an n-absorbing submodule if whenever $ a_ {1}\ cdots a_ {n} m\ in N $ for $ a_ {1},\ ldots, a_ {n}\ in R $ and $ m\ in M ,theneither, then either ,theneither a_ {1}\ cdots a_ {n}\ in (N: _R M) $ or there are $ n-1$ of $ a_ {i} $'s whose product with m is in ...

Abstract In this talk, all rings are commutative with nonzero identity and all modules are consid... more Abstract In this talk, all rings are commutative with nonzero identity and all modules are considered to be unitary. Prime submodules have an important role in the module theory over commutative rings. We recall that a proper submodule N of R-module M is called a prime submodule if whenever a∈ R and m∈ M with am∈ N, then either m∈ N or a∈(N: RM).

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

International Electronic Journal of Algebra, Jan 5, 2021

Second Seminar on Algebra and its Applications

International Electronic Journal of Algebra

Thai Journal of Mathematics, 2012

Kyungpook mathematical journal, 2016

Mathematical Communications

Mathematical Communications, Dec 5, 2012

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .

Thai Journal of Mathematics, 2011

Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce... more Let R be a commutative ring with a nonzero identity and let M be a unitary R-module. We introduce the concepts of 2-absorbing and weakly 2-absorbing submodules of M and give some basic properties of these classes of submodules. Indeed these are generalizations of prime and weak prime submodules. A proper submodule N of M is called a 2-absorbing (resp. weakly 2-absorbing) submodule of M if whenever a, b ∈ R, m ∈ M and abm ∈ N (resp. 0 = abm ∈ N), then ab ∈ (N : R M) or am ∈ N or bm ∈ N. It is shown that the intersection of each distinct pair of prime (resp. weak prime) submodules of M is 2-absorbing (resp. weakly 2-absorbing). We will also show that if R is a commutative ring, M a cyclic R-module and N a 2-absorbing submodule of M , then either (1) M − radN = P is a prime submodule of M such that P 2 ⊆ N or (2) M − radN = P 1 ∩ P 2 , P 1 P 2 ⊆ N and (M − radN) 2 ⊆ N where P 1 , P 2 are the only distinct minimal prime submodules of N .