Narrow Results

We study the primeness of noncommutative polynomials on prime rings. Let $R$ be a prime ring with extended centroid $C$, $\rho$ a right ideal of $R$, $f(X_1,\dots ,X_t)$ a noncommutative polynomial over $C$, which is not a polynomial identity (PI) for $\rho$, and $a,b\in R\setminus \{0\}$. Then $af(x_1,\dots ,x_t)b=0$ for all $x_1,\dots ,x_t\in \rho$ if and only if one of the following holds: (i) $a\rho =0$; (ii) $\rho C=eRC$ for some idempotent $e\in RC$ and $b\in \rho C$ such that either $f(X_1,\dots ,X_t)X_{t+1}$ is a PI for $\rho$ or $f(X_1,\dots ,X_t)$ is central-valued on $eRCe$ and $ab=0$. We then apply the result to higher commutators of right ideals. Some results of the paper are also studied from the view of point of the notion of $X$-primeness of rings.

Essential ideals of multiplicatively semiprime algebras are themselves multiplicatively semiprime algebras and memorize the extended centroid.

An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

Let $R$ be a prime ring with characteristic different from two, $d$ a derivation of $R$, $L$ a noncentral Lie ideal of $R$, and $a\in R$. In the present paper, it is shown that if one of the following conditions holds: (i) $a{(d{(uv)}^s-{(uv)}^t)}^n=0$, (ii) $a{(d{(uv)}^s+{(uv)}^t)}^n=0$, (iii) $a{(d{(uv)}^s-{(vu)}^t)}^n=0$ and (iv) $a{(d{(uv)}^s+{(vu)}^t)}^n=0$ for all $u,v\in L$, where $n,s,t$ are fixed positive integers, then $a=0$ unless $R$ satisfies $s_4$, the standard polynomial identity in four variables.

Let $R$ be a semiprime ring, not necessarily with unity, with extended centroid $C$. For $a\in R$, let $T(a)$ (respectively $I(a)$, $\mathrm{\text{Ref}}(a)$) denote the set of all outer (respectively inner, reflexive) inverses of $a$ in $R$. In the paper, we study the inclusion properties of $T(a)$, $I(a)$ and $\mathrm{\text{Ref}}(a)$. Among other results, we prove that for $a,b\in R$ with $a$ von Neumann regular, $\mathrm{\text{Ref}}(a)\subseteq T(b)$ (respectively $\mathrm{\text{Ref}}(a)\subseteq I(b)$) if and only if $a=\mathrm{\text{E}}[a]b$ (respectively $I(a)\subseteq I(b)$). Here, $\mathrm{\text{E}}[a]$ is the smallest idempotent in $C$ such that $\mathrm{\text{E}}[a]a=a$. This gives a common generalization of several known results.

Let $R$ be a simple algebra over its extended centroid $C$, and let $f(X_1,\dots ,X_n)$ be a noncommutative polynomial having zero constant term. We denote by $f{(R)}^{+}$ the additive subgroup of $R$ generated by all elements $f(x_1,\dots ,x_n)$ for $x_i\in R$. It is proved that if $\mathrm{\text{char}}R=0$, then $f{(R)}^{+}$ is equal to $\{0\}$, $C$, $[R,R]$, or $R$. As to the case $\mathrm{\text{char}}R=p>0$, an example of a polynomial $f$ satisfying $[R,R]⊊f{(R)}^{+}⊊R$ is given. Also, the polynomials $f$ with $f{(R)}^{+}=[R,R]$ are characterized if $\mathrm{\text{char}}R=0$ and $R\ne [R,R]$. Moreover, we work on the context of centrally closed prime algebras to get more general results.

Let R be a prime algebra over a commutative ring K, Z and C the center and extended centroid of R, respectively, g a generalized derivation of R, and f (X₁, …,X_t) a multilinear polynomial over K. If g(f (X₁, …,X_t))ⁿ ∈ Z for all x₁, …, x_t ∈ R, then either there exists an element λ ∈ C such that g(x)= λx for all x ∈ R or f(x₁, …,x_t) is central-valued on R except when R satisfies s₄, the standard identity in four variables.

Let A and B be prime rings and θ: A→ B a bijective additive map such that θ(x) θ(y)=0 for all x, y∈ A with xy=0. Suppose that the maximal right quotient ring Q(A) of A contains a nontrivial idempotent e such that eA ∪ Ae ⊆ A. Then there exists λ ∈ C(B), the extended centroid of B, such that θ(xy) = λ θ(x) θ(y) for all x, y ∈ A. This removes the assumption deg(B)≥ 3 in the recent theorem due to Chebotar, Ke and Lee.

Let R be a prime ring of characteristic different from 2, L a noncentral Lie ideal of R, H and G two nonzero generalized derivations of R. Suppose u^s(H(u)u-uG(u)) u^t=0 for all u ∈ L, where s, t ≥ 0 are fixed integers. Then either (i) there exists p ∈ U such that H(x)=xp for all x ∈ R and G(x)=px for all x ∈ R unless R satisfies S₄, the standard identity in four variables; or (ii) R satisfies S₄ and there exist p, q ∈ U such that H(x)=px+xq for all x ∈ R and G(x)=qx+xp for all x ∈ R.

Let $R$ be a noncommutative prime ring with its Utumi ring of quotients $U$, $C=Z(U)$ the extended centroid of $R$, $G$ a generalized derivation of $R$ and $I$ a nonzero ideal of $R$. If $I$ satisfies any one of the following conditions:

• ⁽ⁱ⁾$[[G({[x,y]}_n)$, ${[x,y]}_n]$, $G({[x,y]}_n)]\in C$,
• ⁽ⁱⁱ⁾$(G(x{\circ }_{n}y)\circ (x{\circ }_{n}y))\circ G(x{\circ }_{n}y)\in C$,

where $n\ge 1$ is a fixed integer, then one of the following holds:

• ⁽¹⁾there exists $\lambda \in C$ such that $G(x)=\lambda x$ for all $x\in R$;
• ⁽²⁾$R$ satisfies $s_4$ and there exist $a\in U$ and $\lambda \in C$ such that $G(x)=ax+xa+\lambda x$ for all $x\in R$;
• ⁽³⁾char $(R)=2$, $R$ satisfies $s_4$ and there exist $\lambda \in C$ and an outer derivation $d$ of $R$ such that $G(x)=\lambda x+d(x)$ for all $x\in R$.

Let $R$ be a prime ring of characteristic not equal to $2$, $U$ be the Utumi quotient ring of $R$ and $C$ be the extended centroid of $R$. Let $G$ and $F$ be two generalized derivations on $R$ and $L$ be a non-central Lie ideal of $R$. If $F(G(u))u=uG(u)$, for all $u\in L$, then one of the following holds :

• ⁽¹⁾$G=0$.
• ⁽²⁾there exists $\lambda \in C$ such that $F(x)=x$, and $G(x)=\lambda x$ for all $x\in R$.
• ⁽³⁾$R$ satisfies the standard identity $s_4$.

Let $R$ be a prime ring of characteristic different from $2$, $U$ its Utumi quotient ring, $C$ its extended centroid, $f(x_1,\dots ,x_n)$ a noncentral multilinear polynomial over $C$, $F$ and $G$ two generalized derivations of $R$ such that $F\ne 0$. We describe all possible forms of $F$ and $G$ in the case

• ⁽¹⁾there exist $\lambda ,\mu \in C$ such that $F(x)=\lambda x$, $G(x)=\mu x$, for any $x\in R$, with $2\mu =1$;
• ⁽²⁾$f{(x_1,\dots ,x_n)}^2$ is central valued on $R$ and there exist $p,u\in U$ and $\lambda ,\mu \in C$ such that $F(x)=[p,x]+\lambda x$, $G(x)=[u,x]+\mu x$ for all $x\in R$, with $2\mu =1$.

Let $R$ be a prime ring of characteristic different from $2$, $Q_r$ be the right Martindale quotient ring of $R$, $C=Z(Q_r)$ the extended centroid of $R$, $L$ be a noncentral Lie ideal of $R$, $H$ and $G$ be two nonzero $b$-generalized derivations of $R$. Suppose there exist fixed integers $m,n\ge 1$ such that $H(u^m)u^n-u^{n}G(u^m)=0$, for all $u\in L$, then either $R$ satisfies the standard identity $s_4(x_1,\dots ,x_4)$ or there is $a^{\prime }\in Q_r$ such that $H(x)=x{a}^{\prime }$, $G(x)=a^{\prime }x$, for any $x\in R$, and one of the following holds:

• ^(a)there exists $a^{\prime }\in C$,
• ^(b)${[x_1,x_2]}^m$ and ${[x_1,x_2]}^n$ are $C$-linearly dependent.

Then, in the second part of the paper we prove a similar result in the case $H$ and $G$ are generalized skew derivations of $R$ such that $H(x^m)x^n-x^{n}G(x^m)=0$, for all $x\in R$.