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ZERO DIVISOR GRAPHS OF SKEW GENERALIZED POWER SERIES RINGS
MOUSSAVI, AHMAD,PAYKAN, KAMAL Korean Mathematical Society 2015 대한수학회논문집 Vol.30 No.4
Let R be a ring, (S,${\leq}$) a strictly ordered monoid and ${\omega}$ : S ${\rightarrow}$ End(R) a monoid homomorphism. The skew generalized power series ring R[[S,${\omega}$]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev-Neumann Laurent series rings. In this paper, we investigate the interplay between the ring-theoretical properties of R[[S,${\omega}$]] and the graph-theoretical properties of its zero-divisor graph ${\Gamma}$(R[[S,${\omega}$]]). Furthermore, we examine the preservation of diameter and girth of the zero-divisor graph under extension to skew generalized power series rings.
ARCHIMEDEAN SKEW GENERALIZED POWER SERIES RINGS
Moussavi, Ahmad,Padashnik, Farzad,Paykan, Kamal Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.2
Let R be a ring, ($S,{\leq}$) a strictly ordered monoid, and ${\omega}:S{\rightarrow}End(R)$ a monoid homomorphism. In [18], Mazurek, and Ziembowski investigated when the skew generalized power series ring $R[[S,{\omega}]]$ is a domain satisfying the ascending chain condition on principal left (resp. right) ideals. Following [18], we obtain necessary and sufficient conditions on R, S and ${\omega}$ such that the skew generalized power series ring $R[[S,{\omega}]]$ is a right or left Archimedean domain. As particular cases of our general results we obtain new theorems on the ring of arithmetical functions and the ring of generalized power series. Our results extend and unify many existing results.
SKEW CYCLIC CODES OVER 𝔽<sub>p</sub> + v𝔽<sub>p</sub> + v<sup>2</sup>𝔽<sub>p</sub>
Mousavi, Hamed,Moussavi, Ahmad,Rahimi, Saeed Korean Mathematical Society 2018 대한수학회보 Vol.55 No.6
In this paper, we study an special type of cyclic codes called skew cyclic codes over the ring ${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$, where p is a prime number. This set of codes are the result of module (or ring) structure of the skew polynomial ring (${\mathbb{F}}_p+v{\mathbb{F}}_p+v^2{\mathbb{F}}_p$)[$x;{\theta}$] where $v^3=1$ and ${\theta}$ is an ${\mathbb{F}}_p$-automorphism such that ${\theta}(v)=v^2$. We show that when n is even, these codes are either principal or generated by two elements. The generator and parity check matrix are proposed. Some examples of linear codes with optimum Hamming distance are also provided.
Skew cyclic codes over $\mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p$
Hamed Mousavi,Ahmad Moussavi,Saeed Rahimi 대한수학회 2018 대한수학회보 Vol.55 No.6
In this paper, we study an special type of cyclic codes called skew cyclic codes over the ring $\mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p$, where $p$ is a prime number. This set of codes are the result of module (or ring) structure of the skew polynomial ring $(\mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p)[x;\theta]$ where $v^3=1$ and $\theta$ is an $\mathbb{F}_p$-automorphism such that $\theta(v)=v^2$. We show that when $n$ is even, these codes are either principal or generated by two elements. The generator and parity check matrix are proposed. Some examples of linear codes with optimum Hamming distance are also provided.
A NOTE ON MINIMAL PRIME IDEALS
Mohammadi, Rasul,Moussavi, Ahmad,Zahiri, Masoome Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4
Let R be a strongly 2-primal ring and I a proper ideal of R. Then there are only finitely many prime ideals minimal over I if and only if for every prime ideal P minimal over I, the ideal $P/{\sqrt{I}}$ of $R/{\sqrt{I}}$ is finitely generated if and only if the ring $R/{\sqrt{I}}$ satisfies the ACC on right annihilators. This result extends "D. D. Anderson, A note on minimal prime ideals, Proc. Amer. Math. Soc. 122 (1994), no. 1, 13-14." to large classes of noncommutative rings. It is also shown that, a 2-primal ring R only has finitely many minimal prime ideals if each minimal prime ideal of R is finitely generated. Examples are provided to illustrate our results.
Rasul Mohammadi,Ahmad Moussavi,Masoome Zahiri Korean Mathematical Society 2023 대한수학회지 Vol.60 No.6
We introduce two subclasses of abelian McCoy rings, so-called π-CN-rings and π-duo rings, and systematically study their fundamental characteristic properties accomplished with relationships among certain classical sorts of rings such as 2-primal rings, bounded rings etc. It is shown that a ring R is π-CN whenever every nilpotent element of index 2 in R is central. These rings naturally generalize the long-known class of CN-rings, introduced by Drazin [9]. It is proved that π-CN-rings are abelian, McCoy and 2-primal. We also show that, π-duo rings are strongly McCoy and abelian and also they are strongly right AB. If R is π-duo, then R[x] has property (A). If R is π-duo and it is either right weakly continuous or every prime ideal of R is maximal, then R has property (A). A π-duo ring R is left perfect if and only if R contains no infinite set of orthogonal idempotents and every left R-module has a maximal submodule. Our achieved results substantially improve many existing results.
Skew power series extensions of $\alpha$-rigid p.p.-rings
Ebrahim Hashemi,Ahmad Moussavi 대한수학회 2004 대한수학회보 Vol.41 No.4
We investigate skew power series of alpha-rigid p.p.-rings,where alpha is an endomorphism of a ring R which is notassumed to be surjective. For an alpha-rigid ring R,R[[x;alpha]] is right p.p., if and only ifR[[x,x^{-1};alpha]] is right p.p., if and only if R is rightp.p. and any countable family of idempotents in R has a join inI(R).
ON ANNIHILATIONS OF IDEALS IN SKEW MONOID RINGS
Mohammadi, Rasul,Moussavi, Ahmad,Zahiri, Masoome Korean Mathematical Society 2016 대한수학회지 Vol.53 No.2
According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and ${\sigma}:M{\rightarrow}End(R)$ a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and ${\sigma}:M{\rightarrow}End(R)$ is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).
SKEW POWER SERIES EXTENSIONS OF α-RIGID P.P.-RINGS
Hashemi, Ebrahim,Moussavi, Ahmad Korean Mathematical Society 2004 대한수학회보 Vol.41 No.4
We investigate skew power series of $\alpha$-rigid p.p.-rings, where $\alpha$ is an endomorphism of a ring R which is not assumed to be surjective. For an $\alpha$-rigid ring R, R[[${\chi};{\alpha}$]] is right p.p., if and only if R[[${\chi},{\chi}^{-1};{\alpha}$]] is right p.p., if and only if R is right p.p. and any countable family of idempotents in R has a join in I(R).