Maximal Ideals in Commutative Rings: Explained and Solved

• norajill
In summary, a maximal ideal in a commutative ring is the largest possible ideal that is not contained in any other ideal. Unlike other ideals, it cannot be extended to a larger ideal in the ring. Maximal ideals are significant in the study of commutative rings, have applications in algebraic number theory and algebraic geometry, and can be multiple in a ring. If a maximal ideal is also a prime ideal, the quotient ring obtained by dividing the original ring by the maximal ideal is a field, known as the maximal ideal theorem.
norajill

itry to soution this question but ican not , because this out my book

Show that a proper ideal I of a commmutative ring R is a maximal ideal iff for any ideal A of R either A subset of I or A+I=R

Why would you not be able to get a solution just because this is from the book?

1. What is a maximal ideal?

A maximal ideal in a commutative ring is an ideal that is not properly contained in any other ideal of the ring. In other words, it is the largest possible ideal in the ring.

2. How are maximal ideals different from other ideals?

Unlike other ideals, a maximal ideal cannot be extended to a larger ideal in the ring. This means that there are no other ideals between the maximal ideal and the ring itself.

3. What is the significance of maximal ideals in commutative rings?

Maximal ideals play a crucial role in the study of commutative rings, as they are closely related to prime ideals. They also have important applications in algebraic number theory and algebraic geometry.

4. Can a commutative ring have more than one maximal ideal?

Yes, a commutative ring can have multiple maximal ideals. In fact, a ring can have infinitely many maximal ideals.

5. How are maximal ideals related to the quotient ring?

If a maximal ideal is also a prime ideal, then the quotient ring obtained by dividing the original ring by the maximal ideal is a field. This is known as the maximal ideal theorem.

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