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On 2-Absorbing and Weakly 2-Absorbing Primary Ideals of a Commutative Semiring
Soheilnia, Fatemeh Department of Mathematics 2016 Kyungpook mathematical journal Vol.56 No.1
Let R be a commutative semiring. The purpose of this note is to investigate the concept of 2-absorbing (resp., weakly 2-absorbing) primary ideals generalizing of 2-absorbing (resp., weakly 2-absorbing) ideals of semirings. A proper ideal I of R said to be a 2-absorbing (resp., weakly 2-absorbing) primary ideal if whenever $a,b,c{\in}R$ such that $abc{\in}I$ (resp., $0{\neq}abc{\in}I$), then either $ab{\in}I$ or $bc{\in}\sqrt{I}$ or $ac{\in}\sqrt{I}$. Moreover, when I is a Q-ideal and P is a k-ideal of R/I with $I{\subseteq}P$, it is shown that if P is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R, then P/I is a 2-absorbing (resp., weakly 2-absorbing) primary ideal of R/I and it is also proved that if I and P/I are weakly 2-absorbing primary ideals, then P is a weakly 2-absorbing primary ideal of R.
On Graded 2-Absorbing and Graded Weakly 2-Absorbing Primary Ideals
Soheilnia, Fatemeh,Darani, Ahmad Yousefian Department of Mathematics 2017 Kyungpook mathematical journal Vol.57 No.4
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.
ON WEAKLY 2-ABSORBING PRIMARY SUBMODULES OF MODULES OVER COMMUTATIVE RINGS
Darani, Ahmad Yousefian,Soheilnia, Fatemeh,Tekir, Unsal,Ulucak, Gulsen Korean Mathematical Society 2017 대한수학회지 Vol.54 No.5
Assume that M is an R-module where R is a commutative ring. A proper submodule N of M is called a weakly 2-absorbing primary submodule of M if $0{\neq}abm{\in}N$ for any $a,b{\in}R$ and $m{\in}M$, then $ab{\in}(N:M)$ or $am{\in}M-rad(N)$ or $bm{\in}M-rad(N)$. In this paper, we extended the concept of weakly 2-absorbing primary ideals of commutative rings to weakly 2-absorbing primary submodules of modules. Among many results, we show that if N is a weakly 2-absorbing primary submodule of M and it satisfies certain condition $0{\neq}I_1I_2K{\subseteq}N$ for some ideals $I_1$, $I_2$ of R and submodule K of M, then $I_1I_2{\subseteq}(N:M)$ or $I_1K{\subseteq}M-rad(N)$ or $I_2K{\subseteq}M-rad(N)$.
ON WEAKLY 2-ABSORBING PRIMARY SUBMODULES OF MODULES OVER COMMUTATIVE RINGS
Ahmad Yousefian Darani,Fatemeh Soheilnia,Unsal Tekir,Gulsen Ulucak 대한수학회 2017 대한수학회지 Vol.54 No.5
Assume that $M$ is an $R$-module where $R$ is a commutative ring. A proper submodule $N$ of $M$ is called a weakly $2$-absorbing primary submodule of $M $ if $0\neq abm\in N$ for any $a,b\in R$ and $m\in M$, then $ab\in (N:M)$ or $am\in M\mbox{-rad}(N)$ or $bm\in M\mbox{-rad}(N).$ In this paper, we extended the concept of weakly $2$-absorbing primary ideals of commutative rings to weakly $2$-absorbing primary submodules of modules. Among many results, we show that if $N$ is a weakly $2$-absorbing primary submodule of $ M$ and it satisfies certain condition $0\neq I_{1}I_{2}K\subseteq N$ for some ideals $I_{1},I_{2}$ of $R$ and submodule $K$ of $M$, then $ I_{1}I_{2}\subseteq (N:M)$ or $I_{1}K\subseteq M\mbox{-rad}(N)$ or $ I_{2}K\subseteq M\mbox{-rad}(N)$.