Let
be a 2-torsion free semiprime ring with the centre
), U be a non-zero ideal and
:
→
be a derivation mapping. Suppose that
admits
(1) a derivation
satisfying one of the following conditions:
(i)
...),
) −
∈
) for all
∈
(ii)
),
) −
∈
) for all
∈
(iii)
,
−
∈
) for all
∈
(iv)
),
) −
∈
) for all
∈
(v)
),
) −
,
∈
) for all
∈
(2) a non-zero derivation
satisfying one of the following conditions:
(i)
(
),
)) −
∈
) for all
∈
(ii)
(
),
)) +
∈
) for all
∈
Then
contains a non-zero central ideal.
Let R be a prime ring with characteristic different from two and S a non-empty subset of R. Suppose that θ, Φ are endomorphisms of R. An additive mapping F : R → R is called a generalized (θ, ...Φ)-derivation (resp. generalized Jordan (θ, Φ)-derivation) on S if there exists a (θ, Φ)-derivation d : R → R such that F(xy) = F(x)θ(y) + Φ(x)d(y) (resp. F(x
2
) = F(x)θ(x) + Φ(x)d(x)), holds for all x, y ∈ S. Suppose that U is a Lie ideal of R such that u
2
∈ U, for all u ∈ U. In the present paper, it is shown if θ is an automorphism of R then every generalized Jordan (θ, Φ)-derivation F on U is a generalized (θ, Φ)-derivation on U.
Let R be a prime ring. Suppose that $\teta,\phi$ 0 are endomorphisms of R. An additive mapping F : R $\rightarrow$ R is called a generalized ($\teta,\phi$)-derivation if there exists a ...($\teta,\phi$)-derivation d : R $\rightarrow$ R such that $F(xy) = F(x)\teta(y) + \phi{x)d{y) holds for all x,y $\in$ R. Let J be a nonzero Jordan ideal of R. In the present paper we begin by proving the following: If F is a generalized ($\teta,\phi$)-derivation on R which acts as a homomorphism or as an anti- homomorphism on J, then either d = 0 or $J \subseteq Z(R)$.
We first establish the commutativity for the semiprime ring satisfying xn, yxr = ±ys x, ym yt for all x, y in R, where m, n, r, s and t are fixed non-negative integers, and further, we investigate ...the commutativity of rings with unity under some additional hypotheses. Moreover, it is also shown that the above result is true for s-unital rings. Also, we provide some counterexamples which show that the hypotheses of our theorems are not altogether superfluous. The results of this paper generalize some of the well-known commutativity theorems for rings which are right s-unital.
Let n > 1, m , t , s be any positive integers, and let R be an associative ring with identity. Suppose x t x n , y = x , y m y s for all x , y in R . If, further, R is n ‐torsion free, then R is ...commutativite. If n ‐torsion freeness of R is replaced by “ m , n are relatively prime,” then R is still commutative. Moreover, example is given to show that the group theoretic analogue of this theorem is not true in general. However, it is true when t = s = 0 and m = n + 1.
Through much shorter proofs, some new communtativity theorems for rings with unity have been obtained. These results either extend or generalize a few well-known theorems. Our method of proof is ...based on an iteration technique.
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