Examples of 'maximal ideal' in a sentence
Meaning of "maximal ideal"
In mathematics, specifically in abstract algebra, a maximal ideal refers to a proper subset of a ring that is not a subset of any other proper ideal. It is the largest possible ideal within a given ring
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- An ideal which cannot be made any larger (by adjoining any element to it) without making it improper (i.e., equal to the whole of the containing algebraic structure).
How to use "maximal ideal" in a sentence
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maximal ideal
Take m to be the maximal ideal of this ring.
A point is closed if and only if is a maximal ideal of.
Every maximal ideal is closed.
Show that the ideal is a maximal ideal of.
Maximal ideal of a valuation.
And a quotient over a maximal ideal is a field.
Every proper ideal of a ring is contained in a maximal ideal.
Any quotient of a ring by a maximal ideal is a simple ring.
The filtration of a local ring by the powers of its maximal ideal.
Let be a maximal ideal of.
Every countable commutative ring has a maximal ideal.
A maximal ideal of a noncommutative ring might not be prime in the commutative sense.
Localized at the maximal ideal m.
Noncommutative rings in which every prime ideal is contained in a unique maximal ideal.
The ideal of is a maximal ideal if and only if the quotient ring is a field.
See also
A ring is said to be local if it has only one maximal ideal.
Prime and maximal ideal and their characterization in terms of the associated quotient ring.
The dual numbers form a local ring since there is a unique maximal ideal.
The maximal ideal of A is principal.
By the above, any maximal ideal is prime.
More generally, let R be a regular local ring with maximal ideal m.
R is a Noetherian local domain whose maximal ideal is principal, and not a field.
Suppose that R is a commutative local ring, with the maximal ideal m.
The corresponding maximal ideal theorems ( MIT ) are often-though not always-stronger than their PIT equivalents.
In a commutative ring with unity, every maximal ideal is a prime ideal.
The maximal ideal of this ring consists precisely of those germs f with f ( 0 ) 0.
Define f ( x ) to be the highest power of the maximal ideal M containing x.
Thus the following ( strong ) maximal ideal theorem ( MIT ) for Boolean algebras is equivalent to BPI,.
As a result, every non-unit of A is contained in a maximal ideal.
For example, formula 1 is a maximal ideal in formula 2, but formula 3 is not a field.
We also write ( R, m ) for a commutative local ring R with maximal ideal m.
Replacing A by a localization, we can assume A is a local ring with maximal ideal p {\displaystyle {\mathfrak {p.
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