The main result: Let R be a 2-torsion free semiprime ring and let T : R → R be an additive mapping. Suppose that T (xyx) = xT (y)x holds for all x, y ∈ R. In this case T is a centralizer.

Let R be a 2-torsion free prime ring with center Z, right Utumi quotient ring U , generalized derivation F associated with a nonzero derivation d of R and L a Lie ideal of R. If F (uv)n = F (u)mF (v)l or F (uv)n = F (v)lF (u)m for all u, v ∈ L, where m,n, l are fixed positive integers, then L ⊆ Z. We also examine the case where R is 312 Ahmad N. Alkenani et al. a semiprime ring. Finally, as an ...

Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called...

This paper is an attempt to prove the following result:Let $n>1$ be an integer and let $mathcal{R}$ be a $n!$-torsion-free ring with the identity element. Suppose that $d, delta, varepsilon$ are additive mappings satisfyingbegin{equation}d(x^n) = sum^{n}_{j=1}x^{n-j}d(x)x^{j-1}+sum^{n-1}_{j=1}sum^{j}_{i=1}x^{n-1-j}Big(delta(x)x^{j-i}varepsilon(x)+varepsilon(x)x^{j-i}delta(x)Big)x^{i-1}quadend{e...

The main purpose of this research is to characterize generalized (left) derivations and Jordan (*,*)-derivations on Banach algebras rings using some functional identities. Let A be a unital semiprime algebra let F,G : ? linear mappings satisfying F(x) =-x2G(x-1) for all x Inv(A), where Inv(A) denotes the set invertible elements A. Then both F G are Another result in regard as follows. n > 1 ...

abstract. let r be a 2-torsion free ring with identity. in this paper, first we prove that any jordan left derivation (hence, any left derivation) on the full matrix ringmn(r) (n  2) is identically zero, and any generalized left derivation on this ring is a right centralizer. next, we show that if r is also a prime ring and n  1, then any jordan left derivation on the ring tn(r) of all n×n up...

The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule struc...

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K . The anti-automorphism g 7→ g of G can be extended linearly to an anti-automorphism a 7→ a∗ of KG . Let S∗(KG) = {x ∈ U(KG) | x∗ = x} be the set of all symmetric units of U(KG) . We consider the following question: for which groups G and commutative rings K it is true that S∗(KG) is a subgroup in U(KG...

Abstract. Let R be a 2-torsion free ring with identity. In this paper, first we prove that any Jordan left derivation (hence, any left derivation) on the full matrix ringMn(R) (n 2) is identically zero, and any generalized left derivation on this ring is a right centralizer. Next, we show that if R is also a prime ring and n 1, then any Jordan left derivation on the ring Tn(R) of all n×n uppe...

Let R be a ring with involution. An additive mapping T : R → R is called a left ∗-centralizer (resp. Jordan left ∗-centralizer) if T (xy) = T (x)y∗ (resp. T (x2) = T (x)x∗) holds for all x, y ∈ R, and a reverse left ∗-centralizer if T (xy) = T (y)x∗ holds for all x, y ∈ R. The purpose of this paper is to solve some functional equations involving Jordan left ∗-centralizers on some appropriate su...