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.

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 with extended centroid C, $\rho$ a non-zero right ideal of R and let $f(X_1,\dots,X_t)$ be a polynomial, having no constant term, over C. Suppose that $f(X_1,\dots,X_t)$ is not central-valued on RC. We denote by $f(\rho)$ the additive subgroup of RC> generated by all elements $f(x_1,\dots,x_t)$ for $x_i\in\rho$. The main goals of this note are to prove two results concerning the extension properties of finiteness conditions as follows.

(I) If $f(\rho)$ spans a non-zero finite-dimensional $C$-subspace of $RC$, then $\dim_CRC$ is finite.

(II) If $f(\rho)\ne0$ and is a finite set, then $R$ itself is a finite ring.

AMS 2000 Mathematics subject classification: Primary 16N60; 16R50

In this paper we characterize maps f: R —> R where R is semiprime, f is additive, and [f(x),f(y)] = [x,y] for all x,y ∊ R. It is shown that f(x) = λx + ξ(x) where λ ∊ C, λ2 = 1, and ξ: R —> C is additive where C is the extended centroid of R.