Can someone please explain the meaning of "maximal ideals" to me ? I am really having a hard time, trying to understand the definition.Also, why is zero a maximal ideal for any field ?Please Help.
A maximal ideal or a ring R is any ideal M (other than the whole ring R) that is not contained in any non-trivial ideal (M or R). Thus to verify an ideal is maximal, it must be shown that if an ideal I contains M, then I must be R or M.
In any field, {0} is an ideal by itself since it is closed under addition and absorbs products. {0} is clearly not the whole field, so it might be an ideal. To verify this we must check that there are no other ideal containing zero that is not the whole ring or {0}. Suppose I contained M and is not {0}. Then there exists another element r in I. But then r*r^-1=1 is in the ideal I since I absorbs products. If 1 is in an ideal, than the ideal I is the whole ring. Thus M is maximal.
An example of something that is not a maximal ideal of the ring of integers (which is not a field) is M=(4). Since I=(2) contains M but is not the integers or M, M is not maximal. Here (4) means the principle ideal generated by 4.