Let be a commutative domain with identity, and Let be the localization of with respect to the set i.e.
which is a subring of the quotient field of Here is an interesting question that we are going to answer in this post: can be local but be not semilocal for some ? In other words:
Question. Does there exist a commutative domain with identity and such that has only one maximal ideal but has infinitely many maximal ideals?
Answer. Yes.
Proof. Consider the following subring of the field of rational functions :
So is just the localization of at the maximal ideal So is local with the unique maximal ideal
Now consider the ring
We show that has infinitely many maximal ideals. Let and define the map by for all
i) is well-defined. That is because if then, since is odd, and so is defined.
ii) is an onto ring homomorphism. Well, it is obvious that is a ring homomorphism. To see that is onto, let where We can write for some integer and some odd integer Now, choose Then and therefore proving that is onto.
So now that we have shown that is an onto ring homomorphism, we get that and so is a maximal ideal of for every integer Finally, if are two nonzero integers, then but and so Hence is an infinite set of maximal ideals of
Note. The example above is a form of an exercise in page 158 of Donald Passman’s book A Course in Ring Theory. My next post is an interesting Lemma in that book which is related to the question we just answered.
Exercise. For a commutative domain with identity and show that Hint. Show that the map defined by is an onto ring homomorphism. To show that write
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