Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Maximal ideal
Summary
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.
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 J = I or J = R.
The quotient ring R/I is a simple ring.
There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R:
There exists no other proper right ideal B of R so that A ⊊ B.
For any right ideal B with A ⊆ B, either B = A or B = R.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.