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Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal in a commutative ring R {\displaystyle R} whenever there exists an integer n > 1 {\displaystyle n>1} such that x n = x {\displaystyle x^{n}=x} for any x ∈ R {\displaystyle x\in R} .
A ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R - module , and P is a maximal element in the poset of annihilator ideals of submodules ...
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.
In Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter. There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint ...
Every prime ideal P of R such that R/P has an element x with (R/P)[x −1] a field is a maximal prime ideal. The spectrum of R is a Jacobson space, meaning that every closed subset is the closure of the set of closed points in it. (For Noetherian rings R): R has no prime ideals P such that R/P is a 1-dimensional semi-local ring.
Thus, points in n-space, thought of as the max spec of = [, …,], correspond precisely to 1-dimensional representations of R, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then ...
If R and S are commutative, M is a maximal ideal of S, and f is surjective, then f −1 (M) is a maximal ideal of R. If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R. If R and S are commutative, S is a field, and f is surjective, then ker(f) is a maximal ideal of R.
the prime ideal m v is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of R v), the residue field k v = R v /m v, the place of K associated to v, the class of v under the equivalence defined below.