I still not understand clearly about group near-ring. Definition of group near-ring given here:
- multiplication in G implies usual multiplication or it can be any defined multiplication?
- does G not necessarily finite since group near-ring NG consist of all finite formal sums?
- why it must be satisfied the four condition?
- this question related to 3rd question, from $\alpha_1g_1(\beta_1h_1+...+\beta_mh_m)$ to $\alpha_1(\beta_1g_1h_1+...+\beta_mg_1h_m)$, why is it possible even the Group G and near-ring N not always commutative? it should be $\alpha_1(g_1\beta_1h_1+...+g_1\beta_mh_m)$, right?
- what is the operation of group near-ring NG since it called group? it depends on the group G and near-ring N?
- Below the word "This will be defined in this text in two ways:", number one should be, $\sum \alpha_1g_1 \sum \beta_jh_j = \sum \alpha_ig_i$ instead of $\sum \alpha_1g_1 \sum \beta_jh_j = \sum \alpha_1g_i$, right?
Thank You so much, forgive me if it's too many question in a single ask.