Maximal left ideals in near-rings of continuous functions on disconnected groups (Q2640683)

The author gives a very nice characterization of the maximal left ideals of the near-rings of all continuous selfmaps of certain topological groups. Let N be a near-ring, let \(\Gamma\) be an N-group and let \(\Delta\) be an ideal of \(\Gamma\). The \(\Delta\)-set, \(\Delta\) (n) of an element \(n\in N\) is defined by \(\Delta (n)=\{\gamma \in \Gamma:\) \(n\gamma\in \Delta \}\). A subset S of N is fixed if \(\cap \{\Delta (n):\) \(n\in S\}\neq \emptyset\). Let \(\Delta\) be an ideal of the N-group \(\Gamma\). Then the ordered pair (\(\Gamma\),\(\Delta\)) is a Hewitt N-group if every finitely generated proper ideal of N is fixed and \(\Delta\) (n)\(\neq \emptyset\) for some \(n\in N\). Now let G be a topological group and let N(G) be the near- ring of all continuous selfmaps of G. The pair (G,\(\mu\)) is an N(G)-group where the mapping \(\mu\) from N(G)\(\times G\) into G is defined by \(\mu (r,x)=f(x)\). Denote the component of 0 in G by \(G_ 0\). It was shown by \textit{R. D. Hofer} [in J. Aust. Math. Soc., Ser. A 28, No.4, 433-451 (1979; Zbl 0431.16015)] that \(G_ 0\) is an ideal of the N(G)-group G. The author shows here that if \((G,G_ 0)\) is a Hewitt N(G)-group, then the fixed maximal left ideals of N(G) are precisely the sets \(L_ c=\{f\in N(G):\) \(f[C]\subseteq G_ 0]\) where C is a component of G. Evidently, these sets are different for different components so that he has exhibited a one-to-one correspondence between the fixed maximal left ideals of the near-ring N(G) and the components of the topological group G. He goes on to show that if \(| G/G_ 0| >2\) and \(G/G_ 0\) is either discrete or compact, then \((G,G_ 0)\) is a Hewitt N(G)-group. He then shows that if \(| G/G_ 0| >2\) and \(G/G_ 0\) is compact, then all proper left ideals of N(G) are fixed and, consequently, the maximal left ideals of N(G) are precisely the sets \(L_ c=\{f\in N(G):\) \(f[C]\subseteq G_ 0]\) where C is a component of G. Finally, he shows that if \(| G/G_ 0| =2\), then there exists no ideal \(\Delta\) of the N(G)-group G such that (G,\(\Delta\)) is a Hewitt N(G)-group.