Let R be a semiprime ring with extended centroid C and let $I(x)$ denote the set of all inner inverses of a regular element x in R. Given two regular elements $a, b$ in R, we characterise the existence of some $c\in R$ such that $I(a)+I(b)=I(c)$ . Precisely, if $a, b, a+b$ are regular elements of R and a and b are parallel summable with the parallel sum ${\cal P}(a, b)$ , then $I(a)+I(b)=I({\cal P}(a, b))$ . Conversely, if $I(a)+I(b)=I(c)$ for some $c\in R$ , then $\mathrm {E}[c]a(a+b)^{-}b$ is invariant for all $(a+b)^{-}\in I(a+b)$ , where $\mathrm {E}[c]$ is the smallest idempotent in C satisfying $c=\mathrm {E}[c]c$ . This extends earlier work of Mitra and Odell for matrix rings over a field and Hartwig for prime regular rings with unity and some recent results proved by Alahmadi et al. [‘Invariance and parallel sums’, Bull. Math. Sci.10(1) (2020), 2050001, 8 pages] concerning the parallel summability of unital prime rings and abelian regular rings.

Bosonizations of quantum linear spaces are a large class of pointed Hopf algebras that include the Taft algebras and their generalizations. We give conditions for the smash product of an associative algebra with a bosonization of a quantum linear space to be (semi)prime. These are then used to determine (semi)primeness of certain smash products with quantum affine spaces. This extends Bergen’s work on Taft algebras.

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

A ring is called right annelidan if the right annihilator of any subset of the ring is comparable with every other right ideal. In this paper we develop the connections between this class of rings and the classes of right Bézout rings and rings whose right ideals form a distributive lattice. We obtain results on localization of right annelidan rings at prime ideals, chain conditions that entail left-right symmetry of the annelidan condition, and construction of completely prime ideals.

Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\,>\,1$ is a positive integer. We characterize $a\,\in \,K$, the Lie algebra of skew elements in $R$, satisfying ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on $K$. This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if $a,\,b\,\in \,R$ satisfy ${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$, where either $n$ is even or $b\,=\,0$, then ${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some $\lambda \,\in \,C$.

We determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.

Let $R$ be a prime ring with extended centroid $\text{C,Q}$ maximal right ring of quotients of $R$, $RC$ central closure of $R$ such that ${{\dim}_{C}}(RC)>4,f({{X}_{1}},...,{{X}_{n}})$ a multilinear polynomial over $C$ that is not central-valued on $R$, and $f(R)$ the set of all evaluations of the multilinear polynomial $f({{X}_{1}},...,{{X}_{n}})$ in $R$. Suppose that $G$ is a nonzero generalized derivation of $R$ such that ${{G}^{2}}(u)u\in C$ for all $u\in f(R)$. Then one of the following conditions holds:

• (i) there exists $a\in \text{Q}$ such that ${{a}^{2}}=0$ and either $G(x)=ax$ for all $x\in R$or $G(x)=xa$ for all $x\in R$;

• (ii) there exists $a\in \text{Q}$ such that $0\ne {{a}^{2}}\in C$ and either $G(x)=ax$ for all $x\in R$ or $G(x)=xa$ for all $x\in R$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$is central-valued on $R$;

• (iii) char $(R)=2$ and one of the following holds:

• (a) there exist $a,b,\in \text{Q}$ such that $G(x)=ax+xb$ for all $x\in R$ and ${{a}^{2}}={{b}^{2}}\in C$;

• (b) there exist $a,b,\in \text{Q}$ such that $G(x)=ax+xb$ for all $x\in R,\,{{a}^{2}},{{b}^{2}}\in C$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$;

• (c) there exist $a\in \text{Q}$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R,{{d}^{2}}=0$ and ${{a}^{2}}+d(a)=0$;

• (d) there exist $a\in \text{Q}$ and an $X$-outer derivation $d$ of $R$ such that $G(x)=ax+d(x)$ for all $x\in R,\,{{d}^{2}}=0,\,{{a}^{2}}+d(a)\in C$ and $f{{({{X}_{1}},...,{{X}_{n}})}^{2}}$ is central-valued on $R$.

Moreover, we characterize the form of nonzero generalized derivations $G$ of $R$ satisfying ${{G}^{2}}(x)=\lambda x$ for all $x\in R$, where $\lambda \in C$.

Let $R$ be a prime ring of characteristic diòerent from $2$, let ${{Q}_{r}}$ be its right Martindale quotient ring, and let $C$ be its extended centroid. Suppose that $F$ is a generalized skew derivation of $R,\,L$ a non-central Lie ideal of $R,\,0\,\ne \,a\,\in \,R,\,m\,\ge \,0$ and $n,\,s\,\ge \,1$ fixed integers. If

$$a{{\left( {{u}^{m}}F\left( u \right){{u}^{n}} \right)}^{s}}\,=\,0$$

for all $u\,\in \,L$, then either $R\,\subseteq \,{{M}_{2}}\left( C \right)$, the ring of $2\,\times \,2$ matrices over $C$, or $m\,=\,0$ and there exists $b\,\in \,{{Q}_{r}}$ such that $F\left( x \right)\,=\,bx$, for any $x\,\in \,R$, with $ab\,=\,0$.

Let R be a ring and M a monoid with twisting f:M × M → U(R) and action ω: M→ Aut(R). We introduce and study the concepts of CM-Armendariz and CM-quasi-Armendariz rings to generalise various Armendariz and quasi-Armendariz properties of rings by working on the context of the crossed product R*M over R. The following results are proved: (1) If M is a u.p.-monoid, then any M-rigid ring R is CM-Armendariz; (2) if I is a reduced ideal of an M-compatible ring R with M a strictly totally ordered monoid, then R/I being CM-Armendariz implies that R is CM-Armendariz; (3) if M is a u.p.-monoid and R is a semiprime ring, then R is CM-quasi-Armendariz. These results generalise and unify many known results on this subject.

In this paper, we characterize Jordan $*$-derivations of a 2-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the following. Let $\delta :\,R\,\to \,R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R\,=\,U\,\oplus \,V$ such that both $U$ and $V$ are invariant under $\delta $. Moreover, $*$ is the identity map of $U$ and $\delta {{|}_{U}}$ is a derivation, and the Jordan $*$-derivation $\delta {{|}_{V}}$ is inner. We also prove the following. Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, char $R\,\ne \,2$, and let $\delta $ be a nonzero Jordan $*$-derivation of $R$. If $\delta $ is an elementary operator of $R$, then ${{\dim}_{C}}\,R\,<\,\infty $ and $\delta $ is inner.

Let $R$ be a semiprime ring with center $Z\left( R \right)$. For $x,\,y\,\in \,R$, we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$. If $\sigma $ is a non-identity automorphism of $R$ such that

1

$$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot \right],\,{{x}^{nk}} \right]\,=\,0$$

for all $x\,\in \,R$, where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$. In particular, when $R$ is a prime ring, $R$ is commutative.

We investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings $R$ and $S$, if $R$ and $S$ are Morita equivalent, then so are the quasi-Baer right ring hulls ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(R)$ and ${{\widehat{\text{Q}}}_{\mathfrak{q}}}\mathfrak{B}(S)$ of $R$ and $S$, respectively. As an application, we prove that if unital ${{C}^{*}}$-algebras $A$ and $B$ are Morita equivalent as rings, then the bounded central closure of $A$ and that of B are strongly Morita equivalent as ${{C}^{*}}$-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring $A[G]$ of a torsion-free Abelian group $G$ over a commutative semiprime quasi-continuous ring $A$. Examples that illustrate and delimit the results of this paper are provided.

McCoy proved that for a right ideal A of S = R[x1, . . ., xk] over a ring R, if rS(A) ≠ 0 then rR(A) ≠ 0. We extend the result to the Ore extensions, the skew monoid rings and the skew power series rings over non-commutative rings and so on.

We prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).

Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

A subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.

More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices. We show that m + n = k2. This implies, for example, that n = k2 − k if and only if there exists a (nonassociative) division algebra over F of dimension k.

We construct a countably infinite family of pairwise non-isomorphic maximal $\mathbb{Q}\left[ X \right]$-orders such that the full 2 by 2 matrix rings over these orders are all isomorphic.

Let $R$ be a non-$\text{GPI}$ prime ring with involution and characteristic $\ne 2,3$. Let $K$ denote the skew elements of $R$, and $C$ denote the extended centroid of $R$. Let $\delta$ be a Lie derivation of $K$ into itself. Then $\delta \,=\,\rho \,+\,\varepsilon$ where $\varepsilon$ is an additive map into the skew elements of the extended centroid of $R$ which is zero on $\left[ K,\,K \right]$, and $\rho$ can be extended to an ordinary derivation of $\left\langle K \right\rangle$ into $RC$, the central closure.

The class of rings studied in this paper properly contains the class of right distributive rings which have at least one completely prime ideal in the Jacobson radical. Amongst other results we study prime and semiprime ideals, right noetherian rings with comparability and prove a structure theorem for rings with comparability. Several examples are also given.

For each $n\ge 4$ we construct a class of examples of a minimal $C$-dependent set of $n$ automorphisms of a prime ring $R$, where $C$ is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that the preceding examples are completely general, whereas for $n=6$ an example is given which fails to enjoy any of the nice properties of the above example.