THE COHN-JORDAN EXTENSION AND SKEW MONOID RINGS OVER A QUASI-BAER RING

HASHEMI EBRAHIM Korean Mathematical Society 2006 대한수학회논문집 Vol.21 No.1

A ring R is called (left principally) quasi-Baer if the left annihilator of every (principal) left ideal of R is generated by an idempotent. Let R be a ring, G be an ordered monoid acting on R by $\beta$ and R be G-compatible. It is shown that R is (left principally) quasi-Baer if and only if skew monoid ring $R_{\beta}[G]$ is (left principally) quasi-Baer. If G is an abelian monoid, then R is (left principally) quasi-Baer if and only if the Cohn-Jordan extension $A(R,\;\beta)$ is (left principally) quasi-Baer if and only if left Ore quotient ring $G^{-1}R_{\beta}[G]$ is (left principally) quasi-Baer.

ON WEAK ARMENDARIZ IDEALS

Hashemi, Ebrahim Korean Mathematical Society 2008 대한수학회논문집 Vol.23 No.3

We introduce weak Armendariz ideals which are a generalization of ideals have the weakly insertion of factors property (or simply weakly IFP) and investigate their properties. Moreover, we prove that, if I is a weak Armendariz ideal of R, then I[x] is a weak Armendariz ideal of R[x]. As a consequence, we show that, R is weak Armendariz if and only if R[x] is a weak Armendariz ring. Also we obtain a generalization of [8] and [9].

On Annihilator ideals of a nearring of skew polynomials over a ring

Ebrahim Hashemi 대한수학회 2007 대한수학회지 Vol.44 No.6

An Alternative Perspective of Near-rings of Polynomials and Power series

Ebrahim Hashemi,Fatemeh Shokuhifar,Abdollah Alhevaz 경북대학교 자연과학대학 수학과 2022 Kyungpook mathematical journal Vol.62 No.3

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ∼ on S defined by letting a ∼ b if and only if annS(a) = annS(b), is an equivalence relation. The compressed zero-divisor graph ΓE(S) of S is the undirected graph whose vertices are the equivalence classes in duced by ∼ on S other than [0]S and [1]S, in which two distinct vertices [a]S and [b]S are adjacent if and only if ab = 0 or ba = 0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R0[x] and the near-ring of the power series R0[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam.

On clean and nil clean elements in skew t.u.p.~monoid rings

Ebrahim Hashemi,Marzieh Yazdanfar 대한수학회 2019 대한수학회보 Vol.56 No.1

Let $R$ be an associative ring with identity, $M$ a t.u.p.~monoid with only one unit and $\omega : M \rightarrow {\rm End}(R)$ a monoid homomorphism. Let $R$ be a reversible, $M$-compatible ring and $\alpha=a_{1}g_{1}+\cdots+a_{n}g_{n}$ a non-zero element in skew monoid ring $R \ast M$. It is proved that if there exists a non-zero element $\beta=b_{1}h_{1}+\cdots+b_{m}h_{m}$ in $R \ast M$ with $\alpha\beta=c$ is a constant, then there exist $1 \leq i_{0} \leq n$, $1 \leq j_{0} \leq m$ such that $g_{i_{0}}=e=h_{j_{0}}$ and $a_{i_{0}}b_{j_{0}}=c$ and there exist elements $a, 0\neq r$ in $R$ with $\alpha r=ca$. As a consequence, it is proved that $\alpha \in R \ast M$ is unit if and only if there exists $1 \leq i_{0} \leq n$ such that $g_{i_{0}}=e, a_{i_{0}}$ is unit and $a_{j}$ is nilpotent for each $j \neq i_{0}$, where $R$ is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of $R$ and those elements in skew monoid ring $R \ast M$, where $R$ is a reversible or right duo ring.

(Σ, ∆)-Compatible Skew PBW Extension Ring

Hashemi, Ebrahim,Khalilnezhad, Khadijeh,Alhevaz, Abdollah Department of Mathematics 2017 Kyungpook mathematical journal Vol.57 No.3

Ever since their introduction, skew PBW ($Poincar{\acute{e}}$-Birkhoff-Witt) extensions of rings have kept growing in importance, as researchers characterized their properties (such as primeness, Krull and Goldie dimension, homological properties, etc.) in terms of intrinsic properties of the base ring, and studied their relations with other fields of mathematics, as for example quantum mechanics theory. Many rings and algebras arising in quantum mechanics can be interpreted as skew PBW extensions. Our aim in this paper is to study skew PBW extensions of Baer, quasi-Baer, principally projective and principally quasi-Baer rings, in the case when the base ring R is not assumed to be reduced. We just impose some mild compatibleness over the base ring R, and prove that these properties are stable over this kind of extensions.

Ore Extension Rings with Constant Products of Elements

Hashemi, Ebrahim,Alhevaz, Abdollah Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.4

Let R be an associative unital ring with an endomorphism α and α-derivation δ. The constant products of elements in Ore extension rings, when the coefficient ring is reversible, is investigated. We show that if f(x) = ∑ⁿ_i=0 a_ixⁱ and g(x) = ∑^m_j=0 b_jx^j be nonzero elements in Ore extension ring R[x; α, δ] such that g(x)f(x) = c ∈ R, then there exist non-zero elements r, a ∈ R such that rf(x) = ac, when R is an (α, δ)-compatible ring which is reversible. Among applications, we give an exact characterization of the unit elements in R[x; α, δ], when the coeficient ring R is (α, δ)-compatible. Furthermore, it is shown that if R is a weakly 2-primal ring which is (α, δ)-compatible, then J(R[x; α, δ]) = N iℓ(R)[x; α, δ]. Some other applications and examples of rings with this property are given, with an emphasis on certain classes of NI rings. As a consequence we obtain generalizations of the many results in the literature. As the final part of the paper we construct examples of rings that explain the limitations of the results obtained and support our main results.

ON CLEAN AND NIL CLEAN ELEMENTS IN SKEW T.U.P. MONOID RINGS

Hashemi, Ebrahim,Yazdanfar, Marzieh Korean Mathematical Society 2019 대한수학회보 Vol.56 No.1

Let R be an associative ring with identity, M a t.u.p. monoid with only one unit and ${\omega}:M{\rightarrow}End(R)$ a monoid homomorphism. Let R be a reversible, M-compatible ring and ${\alpha}=a_1g_1+{\cdots}+a_ng_n$ a non-zero element in skew monoid ring $R{\ast}M$. It is proved that if there exists a non-zero element ${\beta}=b_1h_1+{\cdots}+b_mh_m$ in $R{\ast}M$ with ${\alpha}{\beta}=c$ is a constant, then there exist $1{\leq}i_0{\leq}n$, $1{\leq}j_0{\leq}m$ such that $g_{i_0}=e=h_{j_0}$ and $a_{i_0}b_{j_0}=c$ and there exist elements a, $0{\neq}r$ in R with ${\alpha}r=ca$. As a consequence, it is proved that ${\alpha}{\in}R*M$ is unit if and only if there exists $1{\leq}i_0{\leq}n$ such that $g_{i_0}=e$, $a_{i_0}$ is unit and aj is nilpotent for each $j{\neq}i_0$, where R is a reversible or right duo ring. Furthermore, we determine the relation between clean and nil clean elements of R and those elements in skew monoid ring $R{\ast}M$, where R is a reversible or right duo ring.

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).

ON ANNIHILATOR IDEALS OF A NEARRING OF SKEW POLYNOMIALS OVER A RING

Hashemi, Ebrahim Korean Mathematical Society 2007 대한수학회지 Vol.44 No.6

For a ring endomorphism ${\alpha}$ and an ${\alpha}-derivation\;{\delta}$ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring $R[x;{\alpha},{\delta}]\;and\;R_0[[x;{\alpha}]]$. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its "multiplication" operation.