• Title/Summary/Keyword: Semiprime ring

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SEMIPRIME RINGS WITH INVOLUTION AND CENTRALIZERS

  • ANSARI, ABU ZAID;SHUJAT, FAIZA
    • Journal of applied mathematics & informatics
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    • v.40 no.3_4
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    • pp.709-717
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    • 2022
  • The objective of this research is to prove that an additive mapping T : R → R is a left as well as right centralizer on R if it satisfies any one of the following identities: (i) T(xnyn + ynxn) = T(xn)yn + ynT(xn) (ii) 2T(xnyn) = T(xn)yn + ynT(xn) for each x, y ∈ R, where n ≥ 1 is a fixed integer and R is any n!-torsion free semiprime ring. In addition, we talk over above identities in the setting of *-ring(ring with involution).

ADDITIVE MAPS OF SEMIPRIME RINGS SATISFYING AN ENGEL CONDITION

  • Lee, Tsiu-Kwen;Li, Yu;Tang, Gaohua
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.3
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    • pp.659-668
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    • 2021
  • Let R be a semiprime ring with maximal right ring of quotients Qmr(R), and let n1, n2, …, nk be k fixed positive integers. Suppose that R is (n1+n2+⋯+nk)!-torsion free, and that f : 𝜌 → Qmr(R) is an additive map, where 𝜌 is a nonzero right ideal of R. It is proved that if [[…[f(x), xn1], …], xnk] = 0 for all x ∈ 𝜌, then [f(x), x] = 0 for all x ∈ 𝜌. This gives the result of Beidar et al. [2] for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p = Σki=1 ni and if f : R → Qmr(R) is an additive map satisfying [[…[f(x), xn1], …], xnk] = 0 for all x ∈ R, then [f(x), x] = 0 for all x ∈ R.

ON QUASI-RIGID IDEALS AND RINGS

  • Hong, Chan-Yong;Kim, Nam-Kyun;Kwak, Tai-Keun
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.2
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    • pp.385-399
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    • 2010
  • Let $\sigma$ be an endomorphism and I a $\sigma$-ideal of a ring R. Pearson and Stephenson called I a $\sigma$-semiprime ideal if whenever A is an ideal of R and m is an integer such that $A{\sigma}^t(A)\;{\subseteq}\;I$ for all $t\;{\geq}\;m$, then $A\;{\subseteq}\;I$, where $\sigma$ is an automorphism, and Hong et al. called I a $\sigma$-rigid ideal if $a{\sigma}(a)\;{\in}\;I$ implies a $a\;{\in}\;I$ for $a\;{\in}\;R$. Notice that R is called a $\sigma$-semiprime ring (resp., a $\sigma$-rigid ring) if the zero ideal of R is a $\sigma$-semiprime ideal (resp., a $\sigma$-rigid ideal). Every $\sigma$-rigid ideal is a $\sigma$-semiprime ideal for an automorphism $\sigma$, but the converse does not hold, in general. We, in this paper, introduce the quasi $\sigma$-rigidness of ideals and rings for an automorphism $\sigma$ which is in between the $\sigma$-rigidness and the $\sigma$-semiprimeness, and study their related properties. A number of connections between the quasi $\sigma$-rigidness of a ring R and one of the Ore extension $R[x;\;{\sigma},\;{\delta}]$ of R are also investigated. In particular, R is a (principally) quasi-Baer ring if and only if $R[x;\;{\sigma},\;{\delta}]$ is a (principally) quasi-Baer ring, when R is a quasi $\sigma$-rigid ring.

THE RESULTS CONCERNING JORDAN DERIVATIONS

  • Kim, Byung Do
    • Journal of the Chungcheong Mathematical Society
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    • v.29 no.4
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    • pp.523-530
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    • 2016
  • Let R be a 3!-torsion free semiprime ring, and let $D:R{\rightarrow}R$ be a Jordan derivation on a semiprime ring R. In this case, we show that [D(x), x]D(x) = 0 if and only if D(x)[D(x), x] = 0 for every $x{\in}R$. In particular, let A be a Banach algebra with rad(A). If D is a continuous linear Jordan derivation on A, then we see that $[D(x),x]D(x){\in}rad(A)$ if and only if $[D(x),x]D(x){\in}rad(A)$ for all $x{\in}A$.

On *-bimultipliers, Generalized *-biderivations and Related Mappings

  • Ali, Shakir;Khan, Mohammad Salahuddin
    • Kyungpook Mathematical Journal
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    • v.51 no.3
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    • pp.301-309
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    • 2011
  • In this paper we dene the notions of left *-bimultiplier, *-bimultiplier and generalized *-biderivation, and to prove that if a semiprime *-ring admits a left *-bimultiplier M, then M maps R ${\times}$ R into Z(R). In Section 3, we discuss the applications of theory of *-bimultipliers. Further, it was shown that if a semiprime *-ring R admits a symmetric generalized *-biderivation G : R ${\times}$ R ${\rightarrow}$ R with an associated nonzero symmetric *-biderivation R ${\times}$ R ${\rightarrow}$ R, then G maps R ${\times}$ R into Z(R). As an application, we establish corresponding results in the setting of $C^*$-algebra.

A REMARK ON MULTIPLICATION MODULES

  • Choi, Chang-Woo;Kim, Eun-Sup
    • Bulletin of the Korean Mathematical Society
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    • v.31 no.2
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    • pp.163-165
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    • 1994
  • Modules which satisfy the converse of Schur's lemma have been studied by many authors. In [6], R. Ware proved that a projective module P over a semiprime ring R is irreducible if and only if En $d_{R}$(P) is a division ring. Also, Y. Hirano and J.K. Park proved that a torsionless module M over a semiprime ring R is irreducible if and only if En $d_{R}$(M) is a division ring. In case R is a commutative ring, we obtain the following: An R-module M is irreducible if and only if En $d_{R}$(M) is a division ring and M is a multiplication R-module. Throughout this paper, R is commutative ring with identity and all modules are unital left R-modules. Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for each submodule N of M, there exists and ideal I of R such that N=IM. Cyclic R-modules are multiplication modules. In particular, irreducible R-modules are multiplication modules.dules.

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PRIME RADICALS OF SKEW LAURENT POLYNOMIAL RINGS

  • Han, Jun-Cheol
    • Bulletin of the Korean Mathematical Society
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    • v.42 no.3
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    • pp.477-484
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    • 2005
  • Let R be a ring with an automorphism 17. An ideal [ of R is ($\sigma$-ideal of R if $\sigma$(I).= I. A proper ideal P of R is ($\sigma$-prime ideal of R if P is a $\sigma$-ideal of R and for $\sigma$-ideals I and J of R, IJ $\subseteq$ P implies that I $\subseteq$ P or J $\subseteq$ P. A proper ideal Q of R is $\sigma$-semiprime ideal of Q if Q is a $\sigma$-ideal and for a $\sigma$-ideal I of R, I$^{2}$ $\subseteq$ Q implies that I $\subseteq$ Q. The $\sigma$-prime radical is defined by the intersection of all $\sigma$-prime ideals of R and is denoted by P$_{(R). In this paper, the following results are obtained: (1) For a principal ideal domain R, P$_{(R) is the smallest $\sigma$-semiprime ideal of R; (2) For any ring R with an automorphism $\sigma$ and for a skew Laurent polynomial ring R[x, x$^{-1}$; $\sigma$], the prime radical of R[x, x$^{-1}$; $\sigma$] is equal to P$_{(R)[x, x$^{-1}$; $\sigma$ ].

ON 𝜙-SEMIPRIME SUBMODULES

  • Ebrahimpour, Mahdieh;Mirzaee, Fatemeh
    • Journal of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1099-1108
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    • 2017
  • Let R be a commutative ring with non-zero identity and M be a unitary R-module. Let S(M) be the set of all submodules of M and ${\phi}:S(M){\rightarrow}S(M){\cup}\{{\emptyset}\}$ be a function. We say that a proper submodule P of M is a ${\phi}$-semiprime submodule if $r{\in}R$ and $x{\in}M$ with $r^2x{\in}P{\setminus}{\phi}(P)$ implies that $rx{\in}P$. In this paper, we investigate some properties of this class of sub-modules. Also, some characterizations of ${\phi}$-semiprime submodules are given.