Identifying Maximal and Prime Ideals in the Ring ( mathbb{Z}/12mathbb{Z} )

Understanding maximal and prime ideals is fundamental in ring theory, particularly when working with quotient rings such as ( mathbb{Z}/12mathbb{Z} ). This article will guide you through the steps to identify these ideals in the specified ring, providing a comprehensive overview that aligns with SEO standards and is suitable for Google's indexing criteria.

Step 1: Understanding the Structure of ( mathbb{Z}/12mathbb{Z} )

The ring ( mathbb{Z}/12mathbb{Z} ) is a quotient ring, where the equivalence classes of integers modulo 12 form the elements of the ring. Specifically, the elements of ( mathbb{Z}/12mathbb{Z} ) are the residue classes:

t0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11

This structure helps us to see the internal operations completely within these equivalence classes.

Step 2: Identifying the Ideals of ( mathbb{Z}/12mathbb{Z} )

An ideal in ( mathbb{Z}/12mathbb{Z} ) can be described as the ideal generated by a divisor of 12. The divisors of 12 are 1, 2, 3, 4, 6, and 12. The ideals generated by these divisors are constructed as follows:

t( 0mathbb{Z}/12mathbb{Z} {0} ) t( 1mathbb{Z}/12mathbb{Z} mathbb{Z}/12mathbb{Z} {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} ) t( 2mathbb{Z}/12mathbb{Z} {0, 2, 4, 6, 8, 10} ) t( 3mathbb{Z}/12mathbb{Z} {0, 3, 6, 9} ) t( 4mathbb{Z}/12mathbb{Z} {0, 4, 8} ) t( 6mathbb{Z}/12mathbb{Z} {0, 6} )

Step 3: Finding Maximal Ideals

A maximal ideal ( M ) in a ring ( R ) is an ideal such that there are no other ideals contained between ( M ) and ( R ). In ( mathbb{Z}/12mathbb{Z} ), the maximal ideals correspond to the ideals generated by the prime factors of 12. The prime factorization of 12 is ( 2^2 times 3 ).

The prime ideals of ( mathbb{Z}/12mathbb{Z} ) are generated by the primes 2 and 3, and the maximal ideals are:

t( 2mathbb{Z}/12mathbb{Z} {0, 2, 4, 6, 8, 10} ) is maximal. t( 3mathbb{Z}/12mathbb{Z} {0, 3, 6, 9} ) is also maximal.

Step 4: Finding Prime Ideals

A prime ideal ( P ) in a ring ( R ) is an ideal such that if ( ab in P ) for ( a, b in R ), then either ( a in P ) or ( b in P ).

In ( mathbb{Z}/12mathbb{Z} ), the following ideals are prime:

t( 0mathbb{Z}/12mathbb{Z} {0} ) is a prime ideal because it satisfies the definition of a prime ideal. t( 2mathbb{Z}/12mathbb{Z} {0, 2, 4, 6, 8, 10} ) is prime because if the product of two elements is even, at least one of them must be even. t( 3mathbb{Z}/12mathbb{Z} {0, 3, 6, 9} ) is also prime because if the product of two elements is divisible by 3, at least one of them must be divisible by 3.

Summary of Ideals

After the analysis, we summarize the maximal and prime ideals of ( mathbb{Z}/12mathbb{Z} ) as follows:

Maximal Ideals:

t( 2mathbb{Z}/12mathbb{Z} {0, 2, 4, 6, 8, 10} ) t( 3mathbb{Z}/12mathbb{Z} {0, 3, 6, 9} )

Prime Ideals:

t( 0mathbb{Z}/12mathbb{Z} {0} ) t( 2mathbb{Z}/12mathbb{Z} {0, 2, 4, 6, 8, 10} ) t( 3mathbb{Z}/12mathbb{Z} {0, 3, 6, 9} )

This effectively describes the maximal and prime ideals of ( mathbb{Z}/12mathbb{Z} ) and highlights both the identification and characteristics of these ideals.