GroupNearrings

For a long time, it was unclear how to define "correctly" the concept of a group near-ring $R[G]$ for a general near-ring $R$ with identity and a (multiplicatively written) group $G$. This paper gives such a definition and provides sufficient results which show that this is the "right" way to define $R[G]$. It works as follows. Let $R$ and $G$ be as above, and let $R^G$ be all maps from $G$ to $R$. In $M(R^G)$, the set of all maps from $R^G$ into itself, the author defines maps ${[r,g]}$ for $r\in R$ and $g\in G$ by $({[r,g]}(\mu))(h)=r\mu(hg)$ for $\mu\in R^G$ and $h\in G$. Now $M(R^G)$ is a near-ring with respect to pointwise function multiplication and composition. The group near-ring $R[G]$ is then defined as the sub-near-ring generated by all ${[r,g]}$ for $r\in R$, $g\in G$. If $R$ is a ring, we get the usual definition of a group ring. If $R$ is a d.g. near-ring, one gets the special case of a "group d.g. near-ring", as defined by Meldrum in 1976. Familiar results for group rings extend to group near-rings. The interplay between ideals of $R$ and those of $R{[G]}$ is explored in detail.

Reviewed by Guenter Pilz

In this paper the authors define group near-rings for the general case and lay the foundations for further development of the subject.

Let R be a near-ring with identity 1 and G a group with identity e. $R\sp G$ denotes the cartesian direct sum of $\vert G\vert$ copies of $(R,+)$ indexed by the elements of G. $M(R\sp G)$ is the right near-ring of all mappings of the group $R\sp G$ into itself. For $r\in R$ and $g\in G$ by ${[r,g]}$ we denote the mapping in $M(R\sp G)$ defined by $({[r,g]}(\mu))(h)=r\sb{\mu}(hg)$ for all $\mu \in R\sp G$, $h\in G$. The set $\{[r,g]\vert$ $r\in R$, $g\in G\}$ generates a subnear-ring of $M(R\sp G)$ which will be denoted by $R[G]$ and called the group near-ring constructed from $R$ and $G$. If $R$ happens to be a ring, then $R[G]$ is isomorphic to the standard group-ring constructed from $R$ and $G$. If $R$ is distributively generated, then so is $R[G]$ for every group G. Let $\phi$ ->$R\to S$ be an epimorphism of near-rings and ${\bar \phi}$->$R\sp G\to S\sp G$ be the epimorphism defined by ${\bar\phi}(\mu)(g)= \phi(\mu(g))$ for all $g\in G$. We can define an epimorphism $\phi\sp*->R[G]\to S[G]$ by $\phi\sp*(A)(\sigma)={\bar \phi}(A{\tilde \phi}(\sigma))$ for all $\sigma \in S\sp G$, where ${\tilde \phi}$ is a right inverse of ${\bar \phi}$. Put (Ker $\phi)\sp*=Ker \phi\sp*$. The authors show that the mapping $(\ )\sp*$ is an injection from the set of ideals of $R$ into that of $R[G]$. Moreover, (${\cal A}\cap {\cal B})\sp*={\cal A}\sp*\cap {\cal B}\sp*$ and ${\cal A}\sp*+{\cal B}\sp*\subseteq ({\cal A}+{\cal B})\sp*$. Let ${\cal A}\sp*$ be the ideal generated by the set $\{$ [a,e]$\vert$ $a\in {\cal A}\}$. Then the mapping ( )${}\sp+$ is an injection from the set of ideals of R into that of R[G]. Moreover, ${\cal A}\sp+\subseteq {\cal A}\sp*$, (${\cal A}+{\cal B})\sp+={\cal A}\sp++{\cal B}\sp+$ and (${\cal A}\cap {\cal B})\sp+\subseteq {\cal A}\sp++{\cal B}\sp+$. Denote by $\eta$ the function in $R\sp G$ such that $\eta (e)=1$ and $\eta (g)=0$ if $g\ne e$. For an ideal ${\cal A}$ in $R[G]$, put ${\cal A}\sb*=\{a\in R\vert$ $a=(A\eta)(e)$ for some $A\in {\cal A}\}$. Then ${\cal A}\subseteq ({\cal A}\sb*)\sp*$.

->[ B.Pondel?cek ]

This paper explores the link between the ideals of a near-ring $R$ and two classes of ideals in the group near-ring $R[G]$ \ref[L. R. le Riche, J. D. P. Meldrum and A. P. J. van der Walt, Arch. Math. (Basel) 52 (1989), no. 2, 132--139; MR0985595 (90d:16042)]. Many of the results are analogous to those relating ideals of $R$ to ideals of matrix near-rings over $R$, as found by S. J.\ Abbasi, J. D. P. Meldrum and J. H. Meyer [in Proceedings of the Conference on Near-rings and Near-fields (Oberwolfach, 1989), 3--14, Univ. Duisburg, 1995; per bibl.]. In particular, effective use is again made of word ideals in a near-ring->if $w(x_1,\cdots ,x_n)$ is a word in the free additive group on $\{x_i\}$ then the subgroup of $(R,+)$ generated by $\{w(r_1,\cdots ,r_n)| r_i\in R\}$ is actually an ideal of $R$ when $R$ is distributively generated.

In the final section the authors give a pathological example.

Reviewed by Gordon Mason

This paper builds upon the definitions and ideas presented by L. R. le Riche, J. D. P. Meldrum and A. P. J. van der Walt \ref[Arch. Math. (Basel) 52 (1989), no. 2, 132--139; MR0985595 (90d:16042)]. Let $R$ be a right near-ring with identity $1$, $G$ a multiplicative group with identity $e$ and $R\sp{G}$ the Cartesian direct sum of $|G|$ copies of $(R,+)$ indexed by the elements of $G$. $M(R\sp{G})$ is the right near-ring of all mappings of the group $R\sp{G}$ into itself. For any $r \in R$, $g \in G$, define $ [r,g]$ to be the element of $M(R\sp{G})$ such that $([r,g]\mu)(h) = r\mu(hg)$ for all $\mu\in R\sp{G}$ and $h \in G$. The group near-ring $R[G]$ is the subnear-ring of $M(R\sp{G})$ generated by the $ [r,g]$.

For $\Gamma \subseteq R[G]$, ${\rm id}(\Gamma)$ is constructed recursively by the following six rules->(1) $A \in {\rm id}(\Gamma)$ for all $A \in \Gamma$, (2) if $B, C \in {\rm id}(\Gamma)$ then $B+C \in {\rm id} (\Gamma)$, (3) if $B \in {\rm id}(\Gamma)$ and $X \in R[G]$ then $BX \in {\rm id} (\Gamma)$, (4) if $B \in {\rm id}(\Gamma)$ and $X, Y \in R[G]$ then $X(B+Y) -XY \in {\rm id}(\Gamma)$, (5) if $B \in {\rm id} (\Gamma)$ and $X \in R[G]$ then $X + B - X \in {\rm id}(\Gamma)$, and (6) nothing else is in ${\rm id}(\Gamma)$. For any ideal $I$ of $R$ define the ideals $I\sp{+} = {\rm id}\{[a,e]| a \in I\}$ and $I\sp{*} = (I\sp{G} \colon R\sp{G})$.

Assume that $R$ is zero symmetric. If $I$ is an ideal, properties of $I\sp{*}$ and $I\sp}$ are presented. It is then proved that if $A$ is an ideal of $R$ and $A\sp{$ is nilpotent in $R[G]$ then $A$ is nilpotent in $R$. Furthermore, it is shown that $A$ is (semi)prime in $R$ if and only if $A\sp{*}$ is (semi)prime in $R[G]$.

The distributor series for the near-ring $R$ is defined inductively as $D\sp{0}(R) = R$, $D\sp{i+1}(R) = {\rm Gp}\langle (D\sp{i}(R),D\sp{i}(R))\rangle \sp{R}$. A relationship between the distributor series for $R$ and $R[G]$ is given when $R$ is distributively generated->for all $m \geq 0$, (1) $(D\sp{m}(R))\sp{+} \subseteq D\sp{m}(R[G])$ and (2) $D\sp{m}(R[G]) \subseteq (D\sp{m}(G))\sp{*}$.

Let $H$ be a subgroup of $G$. Define $\delta(H)$ to be the left ideal of $R[G]$ generated by the set $\{[1,h]-[1,e]| h \in H\}$. $\delta(H)$ is studied. It is shown that if $H$ is a normal subgroup of $G$ and $R$ is distributively generated then $\delta(H)$ is a two-sided ideal of $R[G]$. It is also shown that there is a natural relationship between the left ideals in $R$ and the left ideals in the group distributively generated near-ring $R[G]$.

Lastly, it is shown how to associate with each two-sided ideal $J$ of $R[G]$ a subgroup $\Omega J$ of $G$. Six properties are shown to hold for $\Omega J$.

Reviewed by David John

Let $R$ be a right near-ring with identity, $G$ be a nontrivial group, and $R^G$ be the direct sum of $|G|$ copies of $(R,+)$. Then the set of all mappings $f\colon R^G\to R^G$, denoted $M(R^G)$, forms a near-ring under pointwise addition and composition. For all $?\in R^G$ and $h\in G$, let the mapping $ [r,g]\colon R^G\to R^G$ be defined by $([r,g](?))(h)=r?(hg)$. The group near-ring $R[G]$ is the sub-near-ring of $M(R^G)$ generated by $\{[r,g]\in M(R^G) \colon r\in R, g\in G\}$.

Avenues of inquiry for group near-rings are analogous to those done in previous papers for matrix near-rings, and relationships between group near-rings and matrix near-rings are discussed throughout the paper. In particular, ideals $A$ in $R$ are used to construct ideals ${^+A}$ and ${^*A}$ in $R[G]$, and ideals ${\scr A}$ in $R[G]$ are used to construct ideals ${_*{\scr A}}$ in $R$. An ideal ${\scr A}$ of $R[G]$ such that ${^+A}\subset{\scr A}\subset{^*A}$ for some ideal $A$ of $R$ is called an intermediate ideal of $R[G]$. It is shown that for every intermediate ideal ${\scr A}$ of $R[G]$, there exists a unique ideal $A$ of $R$ such that ${^+A}\subset{\scr A}\subset {^*A}$. Moreover, ${\scr A}$ is not equal to ${^+B}$ or ${^*B}$ for any ideal $B$ of $R$. A similar result is true for matrix near-rings.

Every ideal in a matrix near-ring is either intermediate or of the form $A^+$ or $A^*$ for some ideal $A$ of $R$. However, an interesting result is that there is an ideal in $R[G]$ that is neither intermediate nor of the form $^+A$ or $^*A$ for any ideal $A$ of $R$. Such an ideal is called exceptional. The last section includes results on modules over $R[G]$ and Jacobson radicals.

The results presented lay the foundation for further study on ideal structure in group near-rings.

->Reviewed by G. Alan Cannon

For group near-rings $R[G]$ ideals $^+A$, $^*A$ are defined and studied, imitating the usual definitions of $A^+,A^*$ for matrix near-ring $M_n(R)$. There are (surprise!) ideals $I$ of $R[G]$ that are not intermediate and also not of the form $^+A$ or of the form $^*A$: they are called exceptional. In particular the augmentation ideal $\Delta=\text{Id}\langle [1,g]-[1,e]:g\in G\rangle_{R[G]}$ of $R[G]$ is always exceptional. Modules over $R[G]$ and $J_2$-radicals are discussed. Many interesting examples illustrate various situations.

->Giovanni Ferrero (Parma)