http://chineseinput.net/에서 pinyin(병음)방식으로 중국어를 변환할 수 있습니다.
변환된 중국어를 복사하여 사용하시면 됩니다.
- 中文 을 입력하시려면 zhongwen을 입력하시고 space를누르시면됩니다.
- 北京 을 입력하시려면 beijing을 입력하시고 space를 누르시면 됩니다.
RISS 인기검색어
- 검색키워드
- 저자 : Mahmood Behboodi (검색결과 6 건)
- 무료
- 기관 내 무료
- 유료
-
THE ANNIHILATING-IDEAL GRAPH OF A RING
Farid Aliniaeifard,Mahmood Behboodi,Yuanlin Li 대한수학회 2015 대한수학회지 Vol.52 No.6
Let S be a semigroup with 0 and R be a ring with 1. We ex- tend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating- ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph Γ(S), and the other definition yields an undirected graph Γ―(S). It is shown that Γ(S) is not necessar- ily connected, but Γ―(S) is always connected and diam(Γ―(S)) 3. For a ring R define a directed graph APOG(R) to be equal to Γ(IPO(R)), where IPO(R) is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph APOG―(R) to be equal to Γ―(IPO(R)). We show that R is an Artinian (resp., Noetherian) ring if and only if APOG(R) has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that APOG―(R) is a complete graph if and only if either (D(R))2 = 0, R is a direct product of two division rings, or R is a local ring with maximal ideal m such that IPO(R) = {0,m,m2,R}. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings Mn×n(R) where n ≥ 2.
-
THE ANNIHILATING-IDEAL GRAPH OF A RING
ALINIAEIFARD, FARID,BEHBOODI, MAHMOOD,LI, YUANLIN Korean Mathematical Society 2015 대한수학회지 Vol.52 No.6
Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph ${\Gamma}$(S), and the other definition yields an undirected graph ${\overline{\Gamma}}$(S). It is shown that ${\Gamma}$(S) is not necessarily connected, but ${\overline{\Gamma}}$(S) is always connected and diam$({\overline{\Gamma}}(S)){\leq}3$. For a ring R define a directed graph ${\mathbb{APOG}}(R)$ to be equal to ${\Gamma}({\mathbb{IPO}}(R))$, where ${\mathbb{IPO}}(R)$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph ${\overline{\mathbb{APOG}}}(R)$ to be equal to ${\overline{\Gamma}}({\mathbb{IPO}}(R))$. We show that R is an Artinian (resp., Noetherian) ring if and only if ${\mathbb{APOG}}(R)$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that ${\overline{\mathbb{APOG}}}(R)$ is a complete graph if and only if either $(D(R))^2=0,R$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that ${\mathbb{IPO}}(R)=\{0,m,m^2,R\}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $M_{n{\times}n}(R)$ where $n{\geq} 2$.
-
Prime M-ideals, M-prime submodules, M-prime radical and M-Baer's lower nilradical of modules
John Beachy,Mahmood Behboodi,Fayezeh Yazdi 대한수학회 2013 대한수학회지 Vol.50 No.6
Let M be a fixed left R-module. For a left R-module X, weintroduce the notion of M-prime (resp. M-semiprime) submodule of Xsuch that in the case M = R, it coincides with prime (resp. semiprime)submodule of X. Other concepts encountered in the general theory areM-m-system sets, M-n-system sets, M-prime radical and M-Baer’s lowernilradical of modules. Relationships between these concepts and basicproperties are established. In particular, we identify certain submodulesof M, called “primeM-ideals”, that play a role analogous to that of prime(two-sided) ideals in the ring R. Using this definition, we show that if Msatisfies condition H (defined later) and HomR(M,X) ≠ 0 for all mod-ules X in the category σ[M], then there is a one-to-one correspondencebetween isomorphism classes of indecomposable M-injective modules inσ[M] and prime M-ideals of M. Also, we investigate the prime M-ideals,M-prime submodules and M-prime radical of Artinian modules.
-
Modules whose classical prime submodules are intersections of maximal submodules
Marzieh Arabi-Kakavand,Mahmood Behboodi 대한수학회 2014 대한수학회보 Vol.51 No.1
Commutative rings in which every prime ideal is an inter- section of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are clas- sical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Further- more, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.
-
PRIME M-IDEALS, M-PRIME SUBMODULES, M-PRIME RADICAL AND M-BAER'S LOWER NILRADICAL OF MODULES
Beachy, John A.,Behboodi, Mahmood,Yazdi, Faezeh Korean Mathematical Society 2013 대한수학회지 Vol.50 No.6
Let M be a fixed left R-module. For a left R-module X, we introduce the notion of M-prime (resp. M-semiprime) submodule of X such that in the case M=R, it coincides with prime (resp. semiprime) submodule of X. Other concepts encountered in the general theory are M-$m$-system sets, M-$n$-system sets, M-prime radical and M-Baer's lower nilradical of modules. Relationships between these concepts and basic properties are established. In particular, we identify certain submodules of M, called "primeM-ideals", that play a role analogous to that of prime (two-sided) ideals in the ring R. Using this definition, we show that if M satisfies condition H (defined later) and $Hom_R(M,X){\neq}0$ for all modules X in the category ${\sigma}[M]$, then there is a one-to-one correspondence between isomorphism classes of indecomposable M-injective modules in ${\sigma}[M]$ and prime M-ideals of M. Also, we investigate the prime M-ideals, M-prime submodules and M-prime radical of Artinian modules.
-
MODULES WHOSE CLASSICAL PRIME SUBMODULES ARE INTERSECTIONS OF MAXIMAL SUBMODULES
Arabi-Kakavand, Marzieh,Behboodi, Mahmood Korean Mathematical Society 2014 대한수학회보 Vol.51 No.1
Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.
연관 검색어 추천
이 검색어로 많이 본 자료
활용도 높은 자료
