Let ${\mathcal{A}}$ be a unital ring with involution. Assume that ${\mathcal{A}}$ contains a nontrivial symmetric idempotent and ${\it\phi}:{\mathcal{A}}\rightarrow {\mathcal{A}}$ is a nonlinear surjective map. We prove that if ${\it\phi}$ preserves strong skew commutativity, then ${\it\phi}(A)=ZA+f(A)$ for all $A\in {\mathcal{A}}$, where $Z\in {\mathcal{Z}}_{s}({\mathcal{A}})$ satisfies $Z^{2}=I$ and $f$ is a map from ${\mathcal{A}}$ into ${\mathcal{Z}}_{s}({\mathcal{A}})$. Related results concerning nonlinear strong skew commutativity preserving maps on von Neumann algebras are given.

In this paper we prove the following result: let $m,n\geq 1$ be distinct integers, let $R$ be an $mn(m+n)|m-n|$-torsion free semiprime ring and let $D:R\rightarrow R$ be an $(m,n)$-Jordan derivation, that is an additive mapping satisfying the relation $(m+n)D(x^{2})=2mD(x)x+2nxD(x)$ for $x\in R$. Then $D$ is a derivation which maps $R$ into its centre.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a semiprime ring with extended centroid $C$ and with maximal right ring of quotients $Q_{mr}(R)$. Let $d{:}\ R\to Q_{mr}(R)$ be an additive map and $b\in Q_{mr}(R)$. An additive map $\delta {:}\ R\to Q_{mr}(R)$ is called a (left) $b$-generalized derivation with associated map $d$ if $\delta (xy)=\delta (x)y+bxd(y)$ for all $x, y\in R$. This gives a unified viewpoint of derivations, generalized derivations and generalized $\sigma $-derivations with an X-inner automorphism $\sigma $. We give a complete characterization of $b$-generalized derivations of $R$ having nilpotent values of bounded index. This extends several known results in the literature.

Let $K$ be a commutative ring with unity, $R$ an associative $K$-algebra of characteristic different from $2$ with unity element and no nonzero nil right ideal, and $f({x}_{1} , \ldots , {x}_{n} )$ a multilinear polynomial over $K$. Assume that, for all $x\in R$ and for all ${r}_{1} , \ldots , {r}_{n} \in R$ there exist integers $m= m(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ and $k= k(x, {r}_{1} , \ldots , {r}_{n} )\geq 1$ such that $\mathop{[{x}^{m} , f({r}_{1} , \ldots , {r}_{n} )] }\nolimits_{k} = 0$. We prove that: (1) if $\text{char} (R)= 0$ then $f({x}_{1} , \ldots , {x}_{n} )$ is central-valued on $R$; and (2) if $\text{char} (R)= p\gt 2$ and $f({x}_{1} , \ldots , {x}_{n} )$ is not a polynomial identity in $p\times p$ matrices of characteristic $p$, then $R$ satisfies ${s}_{n+ 2} ({x}_{1} , \ldots , {x}_{n+ 2} )$ and for any ${r}_{1} , \ldots , {r}_{n} \in R$ there exists $t= t({r}_{1} , \ldots , {r}_{n} )\geq 1$ such that ${f}^{{p}^{t} } ({r}_{1} , \ldots , {r}_{n} )\in Z(R)$, the center of $R$.

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, I a nonzero right ideal of R. Let f(x1,…,xn) be a noncentral multilinear polynomial over C, m≥1 a fixed integer, a a fixed element of R, g a generalized derivation of R. If ag(f(r1,…,rn))m=0 for all r1,…,rn∈I, then one of the following holds:

1. (1) aI=ag(I)=(0);

2. (2) g(x)=qx, for some q∈U and aqI=0;

3. (3) [f(x1,…,xn),xn+1]xn+2 is an identity for I;

4. (4) g(x)=cx+[q,x] for all x∈R, where c,q∈U such that cI=0 and [q,I]I=0.

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that , where is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.

Let K be a commutative ring with unity, R a prime K-algebra of characteristic different from 2, d and δ non-zero derivations of R, f (x1,…, xn) a multilinear polynomial over K.If

then f(x1,…,xnis central-valued on R.

A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset X of R we have xXy ≠ 0 whenever 0 ≠ x, y ∈ R. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n = 1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the n × n matrix ring over F.