(a) What are the prime and maximal ideals in ℤ / 36 ℤ ? (b) Discuss the general case when 36 is

Step 1:u000ALetu0027s recall some key facts about ideals in quotient rings. If $R$ is a ring and $I$ is

(a) What are the prime and maximal ideals in ℤ / 36 ℤ ? (b)...

Key Concepts

Ideals in Modular Rings

In a ring of the form Z/nZ, every ideal is principal, meaning it can be generated by a single element. There is a one?to?one correspondence between the ideals of Z/nZ and the positive divisors of n. This correspondence follows from the fact that Z is a principal ideal domain and every ideal in Z that contains nZ comes from a divisor of n, thus transferring the divisibility structure of n into the ideal structure of Z/nZ.

Prime Ideals in Modular Rings

A prime ideal in any commutative ring is an ideal P such that if a product ab is in P then at least one of a or b is in P. In the case of Z/nZ, the prime ideals correspond to the ideals that come from prime numbers dividing n. More precisely, an ideal in Z/nZ is prime if and only if the quotient ring Z/nZ divided by that ideal is an integral domain, and via the ideal correspondence, this happens precisely when the corresponding divisor leaves n divided by it as a prime number.

Maximal Ideals in Modular Rings

Maximal ideals are those ideals which are proper and not contained within any other proper ideal; equivalently, if M is a maximal ideal then the quotient ring obtained by modding out by M is a field. In rings like Z/nZ, the maximal ideals also arise from the prime divisors of n. Here, by applying the ideal correspondence, an ideal generated by a divisor d is maximal if and only if n/d is a prime number, making the quotient ring isomorphic to a finite field.

Generalization to Z/rZ

When replacing 36 with an arbitrary integer r > 1, the structure of the ideals in Z/rZ remains governed by the divisibility of r. Every ideal in Z/rZ corresponds uniquely to a divisor of r, and the prime and maximal ideals are those for which the quotient ring is an integral domain or a field, respectively. Specifically, an ideal associated with a divisor d is prime (and maximal) if and only if r/d is a prime number. This establishes a clear link between the factorization of r and the ideal structure of the ring Z/rZ.

(a) What are the prime and maximal ideals in $\mathbb{Z} / 36 \mathbb{Z}$ ? (b) Discuss the general case when 36 is replaced by an integer $r>1$.

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