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Displaying similar documents to “On n-derivations and Relations between Elements rⁿ-r for Some n”

On ( σ , τ ) -derivations in prime rings

Mohammad Ashraf, Nadeem-ur-Rehman (2002)

Archivum Mathematicum

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Let R be a 2-torsion free prime ring and let σ , τ be automorphisms of R . For any x , y R , set [ x , y ] σ , τ = x σ ( y ) - τ ( y ) x . Suppose that d is a ( σ , τ ) -derivation defined on R . In the present paper it is shown that ( i ) if R satisfies [ d ( x ) , x ] σ , τ = 0 , then either d = 0 or R is commutative ( i i ) if I is a nonzero ideal of R such that [ d ( x ) , d ( y ) ] = 0 , for all x , y I , and d commutes with both σ and τ , then either d = 0 or R is commutative. ( i i i ) if I is a nonzero ideal of R such that d ( x y ) = d ( y x ) , for all x , y I , and d commutes with τ , then R is commutative. Finally a related result has been obtain...

Relations between Elements r p l - r and p·1 for a Prime p

Andrzej Prószyński (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

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For any positive power n of a prime p we find a complete set of generating relations between the elements [r] = rⁿ - r and p·1 of a unitary commutative ring.

On quasi n -ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu (2022)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of n -ideals and the class of ( 2 , n ) -ideals. A proper ideal I of R is said to be a quasi n -ideal if I is an n -ideal of R . Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the n -ideals, the quasi primary ideals, the ( 2 , n ) -ideals and the p r -ideals. Moreover, we use the quasi n -ideals to characterize...

More on the strongly 1-absorbing primary ideals of commutative rings

Ali Yassine, Mohammad Javad Nikmehr, Reza Nikandish (2024)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with identity. We study the concept of strongly 1-absorbing primary ideals which is a generalization of n -ideals and a subclass of 1 -absorbing primary ideals. A proper ideal I of R is called strongly 1-absorbing primary if for all nonunit elements a , b , c R such that a b c I , it is either a b I or c 0 . Some properties of strongly 1-absorbing primary ideals are studied. Finally, rings R over which every semi-primary ideal is strongly 1-absorbing primary, and rings R over which...

( δ , 2 ) -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with nonzero identity, let ( ) be the set of all ideals of R and δ : ( ) ( ) an expansion of ideals of R defined by I δ ( I ) . We introduce the concept of ( δ , 2 ) -primary ideals in commutative rings. A proper ideal I of R is called a ( δ , 2 ) -primary ideal if whenever a , b R and a b I , then a 2 I or b 2 δ ( I ) . Our purpose is to extend the concept of 2 -ideals to ( δ , 2 ) -primary ideals of commutative rings. Then we investigate the basic properties of ( δ , 2 ) -primary ideals and also discuss the relations among ( δ , 2 ) -primary, δ -primary...

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let R be a commutative ring with identity. If a ring R is contained in an arbitrary union of rings, then R is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in R , then R contains one of them under various conditions.

Generalized derivations on Lie ideals in prime rings

Basudeb Dhara, Sukhendu Kar, Sachhidananda Mondal (2015)

Czechoslovak Mathematical Journal

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Let R be a prime ring with its Utumi ring of quotients U and extended centroid C . Suppose that F is a generalized derivation of R and L is a noncentral Lie ideal of R such that F ( u ) [ F ( u ) , u ] n = 0 for all u L , where n 1 is a fixed integer. Then one of the following holds: ...

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez (2002)

Studia Mathematica

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In rings Γ M of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence M = ( M l ) l 0 (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in Γ M such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to Γ M , provided Γ M is stable under derivation. By a further development of the method, we obtain the main result of the paper,...

On the mappings 𝒵 A and A in intermediate rings of C ( X )

Mehdi Parsinia (2018)

Commentationes Mathematicae Universitatis Carolinae

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In this article, we investigate new topological descriptions for two well-known mappings 𝒵 A and A defined on intermediate rings A ( X ) of C ( X ) . Using this, coincidence of each two classes of z -ideals, 𝒵 A -ideals and A -ideals of A ( X ) is studied. Moreover, we answer five questions concerning the mapping A raised in [J. Sack, S. Watson, C and C * among intermediate rings, Topology Proc. 43 (2014), 69–82].

On skew derivations as homomorphisms or anti-homomorphisms

Mohd Arif Raza, Nadeem ur Rehman, Shuliang Huang (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let R be a prime ring with center Z and I be a nonzero ideal of R . In this manuscript, we investigate the action of skew derivation ( δ , ϕ ) of R which acts as a homomorphism or an anti-homomorphism on I . Moreover, we provide an example for semiprime case.

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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We say that an ideal I on κ λ is semiproper if the corresponding poset I is semiproper. In this paper we investigate properties of semiproper ideals on κ λ .

On norm closed ideals in L ( p , q )

B. Sari, Th. Schlumprecht, N. Tomczak-Jaegermann, V. G. Troitsky (2007)

Studia Mathematica

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It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for X = p (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of L ( p q ) for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in L ( p , q ) for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in L ( p , q ) , including one that has not been studied before. The proofs use various methods...