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maximal cut - translation to russian
математика
максимальный разрез (сети)
![[[Hasse diagram]] of the set ''P'' of [[divisor]]s of 60, partially ordered by the relation "''x'' divides ''y''". The red subset ''S'' = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.](https://commons.wikimedia.org/wiki/Special:FilePath/Lattice of the divisibility of 60 narrow 1,2,3,4.svg?width=200 "[[Hasse diagram]] of the set ''P'' of [[divisor]]s of 60, partially ordered by the relation \"''x'' divides ''y''\". The red subset ''S'' = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element.")
![fence]] consists of minimal and maximal elements only (Example 3).](https://commons.wikimedia.org/wiki/Special:FilePath/Zigzag poset.svg?width=200 "fence]] consists of minimal and maximal elements only (Example 3).")
математика
минимальный элемент
математика
максимальный элемент
Definition
Wikipedia
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R.
Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.
In noncommutative ring theory, a maximal right ideal is defined analogously as being a maximal element in the poset of proper right ideals, and similarly, a maximal left ideal is defined to be a maximal element of the poset of proper left ideals. 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 Jacobson radical J(R).
It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one sided ideals: for example, in the ring of 2 by 2 square matrices over a field, the zero ideal is a maximal two-sided ideal, but there are many maximal right ideals.
