ForcingLinearity

Let $R$ be a commutative ring with identity and $V$ be a unitary $R$-module. The set $M_R(V)=\{f\colon V\to V\mid f(rv)=rf(v), r\in R,\ v\in V\}$ under function addition and function composition is a 0-symmetric near-ring with identity. It is called the near-ring of homogeneous functions determined by the pair $(V,R)$. Let $\scr C$ be the set of all cyclic submodules of $V$ and let $E_R(V;{\scr C})=\{f\in{\rm End}_RV\mid f(C)\subseteq C, \forall C\in\scr C\}$. The structure of the subring $E_R(V;\scr C)$ of $M_R(V)$ has been studied by several authors \ref[see N. Snashall, Proc. Amer. Math. Soc. 116 (1992), no. 4, 921--927; MR1100664 (93b:13016)]. In this paper the author introduces two more subrings of $M_R(V)$ and studies their structure when $R$ is a PID and $V$ is a finitely generated module.

Reviewed by R. K. Markanda

Let $R$ be a ring with identity and let $V$ be a unital (left) $R$-module. The set $M_{R}(V)=\{f\colon V\to V| f(rv)=rf(v)$ for all $r\in R$ and $v\in V\}$ is a near-ring with respect to function addition and composition. This near-ring is called the near-ring of homogeneous functions on $V$ and it contains the ring ${\rm End}_{R}(V)$ of $R$-endomorphisms of $V.$

The authors consider the problem of how much local linearity is needed on a function $f\in M_{R}(V)$ to force $f\in{\rm End}_{R}(V).$ In particular, a collection $\scr{S}=\{W_{\alpha}\}_{\alpha\in\scr{A}}$ of proper submodules of $V$ is said to force linearity on $V$ if whenever $f\in M_{R}(V)$ and $f$ is linear on each $W_{\alpha}\in\scr{S}$, i.e. $f\in {\rm Hom}_{R}(W_{\alpha},V)$ for all $\alpha\in\scr{A},$ then $f\in{\rm End} _{R}(V).$ The minimum number of proper submodules needed to force linearity is called the forcing linearity number for $V,$ denoted by $f\ln(V).$

In this paper the authors determine this number where $V$ is (1) a vector space over a field; (2) a finitely generated free module over an integral domain, not a field; and (3) a finitely generated free module over a local ring, not an integral domain. As a sample from this well-written paper, we mention: Let $V$ be a vector space over a field $F.$ (i) If $\dim_{F}(V)=1,$ then $f\ln(V)=0.$ (ii) If $\dim_{F}(V)=2,$ then $f\ln(V)=\infty.$ (iii) If $F$ is infinite and $\dim_{F}(V)>1,$ then $f\ln(V)=\infty.$ (iv) If $F$ is finite, say $\left| F\right| =q,$ and $V=F^{m},$ $m\geq3,$ then $f\ln(V)=q+2.$ (v) If $F$ is finite, say $\left| F\right| =q$, and $V$ is infinite-dimensional, then $f\ln(V)=q+2.$

Reviewed by Stefan Veldsman

A mapping $f\colon V\to W$ of unitary left $R$-modules is called linear if (1) $f(v_1 + v_2) = f(v_1) + f(v_2)$ for all $v_1, v_2\in V$, and (2) $f(\alpha v) = \alpha f(v)$ for all $\alpha\in F$ and $v\in V$. In this article, the authors introduce recent developments concerning under what circumstances a mapping satisfying condition (2) would be linear \ref[see C. J. Maxson and J. H. Meyer, J. Algebra 223 (2000), no. 1, 190--207; MR1738259 (2000j:16073)]. The article is well written and easily digested by the non-specialist thanks to the illuminating examples.

Reviewed by Yasuyuki Hirano

Let $R$ be a ring with identity and let $V$ be a unital (left) $R$-module. The set $M_{R}(V)$ of all homogeneous functions, i.e.\ $M_{R} (V)=\{f\colon V\rightarrow V\mid f(rv)=rf(v)$ for all $r\in R$ and $v\in V\}$ contains ${\rm End}_{R} (V),$ the set of $R$-endomorphisms of $V.$ In general this inclusion is strict, but there are many classes of rings for which the equality holds.

Earlier the authors introduced a measure of how much local linearity is needed on a function $f\in M_{R}(V)$ to force $f\in {\rm End}_{R} (V)$ \ref[cf. J. Algebra 223 (2000), no. 1, 190--207; MR1738259 (2000j:16073)]. This measure can be quantified as a number (nonnegative or $\infty)$ and is called the forcing linearity number of $V$. It is based on the minimum number of submodules of $V$ on which $f$ is linear. Subsequently this number has been determined for various classes of rings.

In this paper, the authors continue these investigations and they determine the forcing linearity number for modules over simple domains. As an application, the forcing linearity numbers for all finitely generated modules over the first Weyl algebra are determined.

Reviewed by Stefan Veldsman

Let $R$ be a ring with identity and $V$ a unitary $R$-module. A map $f \colon V\to V$ is called $R$-homogeneous provided that $f(rm)=rf(m)$ for any $r\in R$ and $m\in V$. Let $M_{R}(V)$ be the set of all $R$-homogeneous maps on $V$. In the paper under review the author gives a survey of what is currently known for the following questions: $(\alpha)$ When is $M_{R}(V)$ a ring? $(\beta)$ When does $M_{R}(V)$ coincide with the ring of $R$-linear endomorphisms of $V$? $(\gamma)$ Given that $f$ in $M_{R}(V)$ is linear restricted to some prescribed set of proper submodules of $V$, when does it follows that $f$ is $R$-linear? Several classes of rings for which these questions have been answered are described, and, in connection to question $(\gamma)$, there is a brief discussion of the notion of forcing linearity number of a module. In particular this number is computed for certain modules over various classes of rings.

Reviewed by Apostolos D. Beligiannis