Commutative Ring with Nonzero Prime Ideal P = P2: Example and Proof

In summary, a ring is a mathematical structure with two binary operations, addition and multiplication, that forms an abelian group and satisfies certain properties. An integral domain is a commutative ring without zero divisors, meaning every non-zero element has a multiplicative inverse. The main difference between a ring and an integral domain is the presence of zero divisors. Examples of rings and integral domains include integers, rational numbers, and polynomials. Rings and integral domains are important in mathematics as they provide a framework for studying algebraic structures and have applications in various scientific fields.
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Homework Statement


Give an example of a commutative ring R with a 1 and nonzero prime ideal P of R such that P = P2


Homework Equations





The Attempt at a Solution


Letting R be an integral domain and look at the ideal 0xR in RxR. is all i got but not sure how to show this or what to do next
 
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  • #2
Well, you need to show first that 0xR is a prime ideal, and then show that P2=P. How would you approach the first question, that is, showing that it is a prime ideal??
 

What is a ring?

A ring is a mathematical structure consisting of a set of elements, together with two binary operations called addition and multiplication. The addition operation makes the set into an abelian group, while the multiplication operation is associative, distributive over addition, and has an identity element.

What is an integral domain?

An integral domain is a commutative ring in which the product of any two non-zero elements is also non-zero. This means that there are no zero divisors in an integral domain, and therefore every non-zero element has a multiplicative inverse.

What is the difference between a ring and an integral domain?

The main difference between a ring and an integral domain is the presence of zero divisors. In a ring, it is possible for two non-zero elements to multiply to give zero, while in an integral domain, this is not possible. Additionally, integral domains are commutative, while rings may or may not be commutative.

What are some examples of rings and integral domains?

Examples of rings include the set of integers, the set of polynomials with coefficients in a field, and the set of n x n matrices with entries from a ring. Examples of integral domains include the set of integers, the set of rational numbers, and the set of polynomials with coefficients in a field.

Why are rings and integral domains important in mathematics?

Rings and integral domains are important in mathematics because they provide a framework for studying algebraic structures and their properties. They are used extensively in abstract algebra, number theory, and algebraic geometry, and have applications in various scientific fields such as physics and computer science.

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