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Since a one-sided maximal ideal A is not necessarily two-sided, the quotient R/A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the ...
For an example more geometrical in nature, take the ring R = {f/g : f, g polynomials in R[X] and g(0) ≠ 0}, considered as a subring of the field of rational functions R(X) in the variable X. R can be identified with the ring of all real-valued rational functions defined (i.e. finite) in a neighborhood of 0 on the real axis (with the ...
Still more generally, if A is a regular local ring, then the formal power series ring A[[x]] is regular local. If Z is the ring of integers and X is an indeterminate, the ring Z[X] (2, X) (i.e. the ring Z[X] localized in the prime ideal (2, X) ) is an example of a 2-dimensional regular local ring which does not contain a field.
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Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I. The statement of the original theorem can be obtained by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result. To prove the "stronger" result directly, consider the set S of all proper ideals of ...
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Consider a formal group F(X,Y) with coefficients in the ring of integers in a local field (for example Z p). Taking X and Y to be in the unique maximal ideal gives us a convergent power series and in this case we define F(X,Y) = X + F Y and we have a genuine group law. For example if F(X,Y)=X+Y, then this is the usual addition