Let $V$ be a vector space and let $T(V)$ denote the semigroup (under composition) of all linear transformations from $V$ into $V$. For a fixed subspace $W$ of $V$, let $T(V,W)$ be the semigroup consisting of all linear transformations from $V$ into $W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that

$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$

$T(V,W)$

$T(V,W)$

$Q$

$W$

$V$

$Q$

$W$

$n$

$m$

$V$

$F$

The paper presents structural synthesis of maximally regular T3R2-type parallel robotic manipulators (PMs) with five degrees of freedom. The moving platform has three independent translations (T3) and two rotations (R2). A method is proposed for structural synthesis of maximally regular T3R2-type PMs based on the theory of linear transformations and evolutionary morphology. A one-to-one correspondence exists between the actuated joint velocity space and the external velocity space of the moving platform. The Jacobian matrix mapping the two vector spaces of maximally regular T3R2-type PMs presented in this paper is the 5×5 identity matrix throughout the entire workspace. The condition number and the determinant of the Jacobian matrix being equal to one, the manipulator performs very well with regard to force and motion transmission capabilities. Kinematic analysis of maximally regular parallel robots is trivial and no computation is required for real-time control. This paper presents in a unified approach the structural synthesis of PMs with five degrees of freedom with decoupled and uncoupled motions, along with the maximally regular solutions.

If $C=C\left( R \right)$ denotes the center of a ring $R$ and $g\left( x \right)$ is a polynomial in $C\left[ x \right]$, Camillo and Simón called a ring $g\left( x \right)$-clean if every element is the sum of a unit and a root of $g\left( x \right)$. If $V$ is a vector space of countable dimension over a division ring $D$, they showed that $\text{en}{{\text{d}}_{\,D}}V$ is $g\left( x \right)$-clean provided that $g\left( x \right)$ has two roots in $C\left( D \right)$. If $g\left( x \right)=x-{{x}^{2}}$ this shows that $\text{en}{{\text{d}}_{\,D}}V$ is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that $\text{en}{{\text{d}}_{\,R}}M$ is $g\left( x \right)$-clean for any semisimple module $M$ over an arbitrary ring $R$ provided that $g\left( x \right)\in \left( x-a \right)\left( x-b \right)C\left[ x \right] $ where $a,b\in C$ and both $b$ and $b-a$ are units in $R$.