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On rings close to regular and p -injectivity

Roger Yue Chi Ming (2006)

Commentationes Mathematicae Universitatis Carolinae

The following results are proved for a ring A : (1) If A is a fully right idempotent ring having a classical left quotient ring Q which is right quasi-duo, then Q is a strongly regular ring; (2) A has a classical left quotient ring Q which is a finite direct sum of division rings iff A is a left TC -ring having a reduced maximal right ideal and satisfying the maximum condition on left annihilators; (3) Let A have the following properties: (a) each maximal left ideal of A is either a two-sided ideal...

On rings with a unique proper essential right ideal

O. A. S. Karamzadeh, M. Motamedi, S. M. Shahrtash (2004)

Fundamenta Mathematicae

Right ue-rings (rings with the property of the title, i.e., with the maximality of the right socle) are investigated. It is shown that a semiprime ring R is a right ue-ring if and only if R is a regular V-ring with the socle being a maximal right ideal, and if and only if the intrinsic topology of R is non-discrete Hausdorff and dense proper right ideals are semisimple. It is proved that if R is a right self-injective right ue-ring (local right ue-ring), then R is never semiprime and is Artin semisimple...

On skew derivations as homomorphisms or anti-homomorphisms

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

Commentationes Mathematicae Universitatis Carolinae

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.

On some Results Related to Köthe's Conjecture

Agata, Smoktunowicz (2001)

Serdica Mathematical Journal

The Köthe conjecture states that if a ring R has no nonzero nil ideals then R has no nonzero nil one-sided ideals. Although for more than 70 years significant progress has been made, it is still open in general. In this paper we survey some results related to the Köthe conjecture as well as some equivalent problems.

On the characterization of certain additive maps in prime * -rings

Mohammad Ashraf, Mohammad Aslam Siddeeque, Abbas Hussain Shikeh (2024)

Czechoslovak Mathematical Journal

Let 𝒜 be a noncommutative prime ring equipped with an involution ‘ * ’, and let 𝒬 m s ( 𝒜 ) be the maximal symmetric ring of quotients of 𝒜 . Consider the additive maps ℋ and 𝒯 : 𝒜 → 𝒬 m s ( 𝒜 ) . We prove the following under some inevitable torsion restrictions. (a) If m and n are fixed positive integers such that ( m + n ) 𝒯 ( a 2 ) = m 𝒯 ( a ) a * + n a 𝒯 ( a ) for all a ∈ 𝒜 and ( m + n ) ℋ ( a 2 ) = m ℋ ( a ) a * + n a 𝒯 ( a ) for all a ∈ 𝒜 , then ℋ = 0 . (b) If 𝒯 ( a b a ) = a 𝒯 ( b ) a * for all a , b ∈ 𝒜 , then 𝒯 = 0 . Furthermore, we characterize Jordan left τ -centralizers in semiprime rings admitting an anti-automorphism τ . As applications, we find the structure of...

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