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There are other equivalent ways of expressing the definition of maximal one-sided and maximal two-sided ideals. Given a ring R and a proper ideal I of R (that is I ≠ R), I is a maximal ideal of R if any of the following equivalent conditions hold: There exists no other proper ideal J of R so that I ⊊ J. For any ideal J with I ⊆ J, either ...
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 ...
In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal I = m {\displaystyle I={\mathfrak {m}}} is especially important, for example the distinguished maximal ideal of a valuation ring .
Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings. [12] Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
R has a unique maximal left ideal. R has a unique maximal right ideal. 1 ≠ 0 and the sum of any two non-units in R is a non-unit. 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit. If a finite sum is a unit, then it has a term that is a unit (this says in particular that the empty sum cannot be a unit, so it implies 1 ≠ 0).
A subset I of a partially ordered set (,) is an ideal, if the following conditions hold: [1] [2] I is non-empty, for every x in I and y in P, y ≤ x implies that y is in I (I is a lower set), for every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z (I is a directed set).
Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator).
The ideal A is a prime ideal in R if all but one of the A i are equal to R i and the remaining A i is a prime ideal in R i. However, the converse is not true when I is infinite. For example, the direct sum of the R i form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a ...