(4) in Z

Definition

(4) in Z refers to the ideal generated by the integer 4 within the ring of integers, denoted as Z. This ideal consists of all integer multiples of 4, making it a primary ideal since it can be characterized by a single element and exhibits certain properties essential to understanding the structure of ideals in commutative algebra. The concept helps in examining how such ideals interact with the ring, including their intersection and relationship with other ideals.

5 Must Know Facts For Your Next Test

1. (4) in Z is a primary ideal because it can be expressed as the product of a prime ideal (2) in Z.
2. The elements of the ideal (4) in Z include ..., -8, -4, 0, 4, 8, ..., which are all multiples of 4.
3. In any commutative ring, the primary ideals play an essential role in studying the decomposition of ideals into prime ideals.
4. The quotient ring Z/(4) is isomorphic to Z/4Z, which consists of the equivalence classes {0, 1, 2, 3} under addition modulo 4.
5. Any two ideals containing (4) will intersect at least at (4), showcasing how this ideal operates within the larger structure of the integer ring.

Review Questions

• How does the ideal (4) in Z exemplify the properties of primary ideals?
  • (4) in Z demonstrates properties of primary ideals since any product ab belonging to this ideal implies that at least one of the factors a must be part of the ideal or b raised to some power must belong to it. For instance, if we consider any integers a and b where ab is divisible by 4, then either a is itself divisible by 4 or some power of b will lead to that divisibility. This highlights how primary ideals extend the understanding of factorization within rings.
• What role does (4) play when examining the quotient ring Z/(4)?
  • (4) in Z is fundamental when looking at the quotient ring Z/(4) because it creates a structure that encapsulates equivalence classes under addition modulo 4. This means every integer can be grouped into one of four classes based on its remainder when divided by 4: {0, 1, 2, 3}. This quotient not only simplifies calculations but also provides insights into how ideals partition the integer ring into manageable subsets.
• Evaluate the impact of (4) being primary on its interaction with other ideals in Z.
  • The primary nature of (4) significantly affects its interaction with other ideals within Z. Because it can be generated from a prime ideal (2), any intersection or combination involving (4) with another ideal will retain specific characteristics inherent to primary ideals. For example, if you intersect (4) with another ideal like (6), you end up with their greatest common divisor. This emphasizes how understanding primary ideals informs us about more complex relationships between ideals and contributes to overall ring theory.

Related terms

Primary Ideal: An ideal I in a ring R is primary if whenever ab ∈ I for some elements a and b in R, either a ∈ I or b^n ∈ I for some integer n > 0.

Prime Ideal:

A prime ideal P in a ring R is an ideal such that if ab ∈ P for some elements a and b in R, then either a ∈ P or b ∈ P.

Multiplicative Closure: The set of all multiples of an element in a ring, which forms an ideal when considering any element from the ring.