Prime Ideal in a Commutative Ring - Rotman Proposition 7.5

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In summary, Rotman is discussing the proof of Proposition 7.5 and how 1 + I \neq 0 + I in R/I if I is a prime ideal.
  • #1
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I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ...

I am currently studying Section 7.1 Prime Ideals and Maximal Ideals ... ...

I need help with understanding an aspect of the proof of Proposition 7.5

Proposition 7.5 and its proof reads as follows:View attachment 4727In the first part of the proof of the proposition above we read the following:

"Let \(\displaystyle I\) be a prime ideal. Since \(\displaystyle I\) is a proper idea, we have \(\displaystyle 1 \notin I\) and so \(\displaystyle 1 + I \neq 0 + I\) in \(\displaystyle R/I\) ... ... ... "

My question is ... ... why is Rotman taking trouble to show that \(\displaystyle 1 + I \neq 0 + I\) in \(\displaystyle R/I\)?

What is the point Rotman is making ... ... ?

Hope someone can clarify this matter ... ...

Peter
 
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  • #2
Hi Peter,

He is just avoiding the case when $I$ is not proper and then $R/I\cong \{0\}$.
 
  • #3
Fallen Angel has answered the question already, but I will elaborate a bit more on this in case it's not clear :

A proper ideal of a ring $R$ is an ideal which is not the whole ring $R$. By definition, prime ideals are proper. If a prime ideal $I \subset R$ contained the identity $1$, then it has to contain every element of $R$ by definition of an ideal, hence forcing it to be not proper - contradiction.

Thus, $1 \notin I$, which in turn implies $0 \mod I \neq 1 \mod I$ (note : I use $a \mod I$ to denote an element of $R/I$ instead of $a + I$).
 
  • #4
mathbalarka said:
Fallen Angel has answered the question already, but I will elaborate a bit more on this in case it's not clear :

A proper ideal of a ring $R$ is an ideal which is not the whole ring $R$. By definition, prime ideals are proper. If a prime ideal $I \subset R$ contained the identity $1$, then it has to contain every element of $R$ by definition of an ideal, hence forcing it to be not proper - contradiction.

Thus, $1 \notin I$, which in turn implies $0 \mod I \neq 1 \mod I$ (note : I use $a \mod I$ to denote an element of $R/I$ instead of $a + I$).
Fallen Angel, Mathbalarka

Thanks for your help ... appreciate you assistance ...

Peter
 

Related to Prime Ideal in a Commutative Ring - Rotman Proposition 7.5

1. What is a prime ideal in a commutative ring?

A prime ideal in a commutative ring is a subset of the ring that satisfies certain properties. Specifically, it is a subset that is closed under addition and multiplication, contains the multiplicative identity element, and has the property that if the product of two elements is in the subset, then at least one of the elements must also be in the subset. This definition may seem complex, but it essentially means that a prime ideal is a subset of a commutative ring that behaves similarly to a prime number in the integers.

2. How is a prime ideal different from a regular ideal?

A regular ideal is a subset of a ring that is closed under addition and multiplication, contains the multiplicative identity element, and has the property that any element of the ring multiplied by an element of the ideal is also in the ideal. The key difference between a prime ideal and a regular ideal is that a prime ideal has the additional property that if the product of two elements is in the subset, then at least one of the elements must also be in the subset. This extra property is what makes prime ideals important in the study of commutative rings.

3. Can you give an example of a prime ideal in a commutative ring?

Yes, an example of a prime ideal in a commutative ring is the subset {2, 3} in the ring of integers. This subset is closed under addition and multiplication, contains the identity element 1, and has the property that if the product of two elements is in the subset, then at least one of the elements must also be in the subset. In this case, 2 x 3 = 6 is in the subset, and both 2 and 3 are also in the subset.

4. How are prime ideals used in mathematics?

Prime ideals are used in various areas of mathematics, particularly in the study of commutative rings and algebraic geometry. They have important applications in number theory, where they are used to factorize integers, and in algebraic geometry, where they are used to study the structure of algebraic varieties. Prime ideals also have connections to other mathematical concepts, such as prime numbers and irreducible polynomials.

5. What is the significance of Proposition 7.5 in Rotman's book?

Proposition 7.5 in Rotman's book is significant because it provides a characterization of prime ideals in a commutative ring. This proposition states that an ideal in a commutative ring is prime if and only if the quotient ring obtained by dividing the ring by the ideal is an integral domain. This result is important because it allows us to determine whether a given ideal is prime or not by examining the properties of the quotient ring, rather than directly examining the ideal itself.

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