Hey there,
Let R be the ring consisting of the power series of the form
where and are real numbers so that .
I want to show that R is an integral domain without a maximal ideal. But I don't see how to start...
for with and define clearly and so is an integral domain (without the identity element!)
now suppose that is a maximal ideal of let and choose let see that is an ideal of since we have
so is a "proper ideal" of it's clear that because let if then clearly so we'll assume that suppose that then
and therefore which is a false result. so we must have and so thus hence which contradicts maximality of
if is the first non-zero element in the sequence then (and not ) is called the degree of of course we may always "assume" that and so
the notations are standard and i don't know how you didn't understand them?? is the set of real numbers and
2. I dont understand the notation, what does and what does or mean?
also and
that was a terrible typo i made. i just fixed it and you should take a look at it again.
well, if you allow then you'll exactly get the ring which i already mentioned. in this case (your original ring) would be the unique maximal ideal of this new ring
If we allow then R has an identity and therefore a maximal ideal. Prove that in this case the maximal ideal is unique and identify it.
I cant find the maximal ideal, anyone has an idea?
first see that is indeed an ideal of now let be any ideal of not contained in so has an element of the form but then
would have an inverse in and, since is an ideal, we get thus so every proper ideal of is contained in and hence is the unique maximal ideal of