Question

Let \(R\) be an integral domain. Show that $$ a \sim b \Longleftrightarrow(a \mid b \text { and } b \mid a) \Longleftrightarrow\langle a\rangle=\langle b\rangle, $$ where \(\langle a\rangle=R a=\\{r a: r \in R\\}\) is the ideal generated by \(a\).

Short answer

The condition \(a \mid b\) and \(b \mid a\) is equivalent to \(\langle a \rangle = \langle b \rangle\) in an integral domain.

Step-by-step solution

Understand the Notation

In this problem, we are asked to show an equivalence of three conditions about elements \(a\) and \(b\) in an integral domain \(R\). The notation \(a \sim b\) means \(a\) and \(b\) are associates, or each divides the other. Additionally, \(\langle a \rangle = \langle b \rangle\) signifies that the ideals generated by \(a\) and \(b\) are identical.

Show that Divisibility Implies Equal Ideals

Assume \(a \mid b\) and \(b \mid a\). This means there exist elements \(r, s \in R\) such that \(b = ra\) and \(a = sb\). Consider the ideal \(\langle a \rangle\), which contains elements of the form \(ra\). Since \(b = ra\), \(b \in \langle a \rangle\). Similarly, because \(a = sb\), we have \(a \in \langle b \rangle\). Hence, every element in \(\langle a \rangle\) is in \(\langle b \rangle\) and vice versa, so \(\langle a \rangle = \langle b \rangle\).

Show that Equal Ideals Imply Divisibility

Now assume \(\langle a \rangle = \langle b \rangle\). This implies \(a = rb\) for some \(r \in R\) and \(b = sa\) for some \(s \in R\), i.e., \(a \mid b\) and \(b \mid a\). Thus, if the ideals are the same, \(a\) and \(b\) are associates, satisfying \(a \mid b\) and \(b \mid a\).

Conclude the Proof

Since we have shown both directions of implication, \(a \sim b \Longleftrightarrow \langle a \rangle = \langle b \rangle\). Therefore, the statement is proved.

Key concepts

Associates

In the context of an integral domain, two elements are called **associates** if they can be expressed as multiples of each other by a unit of the ring. A unit is an element that has a multiplicative inverse.It's like saying two elements are the same up to multiplication by a non-zero divisor, which in the case of numbers, could be factors like 1 or -1. So, for elements in an integral domain:

• If you can find a unit, say \(u\), such that \(a = ub\), then \(a\) and \(b\) are associates.
• This is similar to saying they're essentially the 'same' in how they contribute to the structure of the ring.

Thus, in the mathematical notation \(a \sim b\), it informs you that "a" and "b" are associated in this way within the integral domain.

Divisibility

Divisibility in integral domains can be akin to our common understanding of dividing numbers. An element \(a\) divides another element \(b\) if there is some element \(r\) in the domain such that \(b = ra\).It means that \(b\) is a scaled or "stretched" version of \(a\). Importantly:

• If \(a \mid b\), then there is no remainder when \(b\) is 'divided' by \(a\).
• The reverse, \(b \mid a\), also holds in this associative virtue, which results in them being multiples of each other, just scaled differently.

This concept becomes critical in determining when two elements are associates, as it rephrases to each element being a divisor of the other.

Ideal

The term "ideal" in ring theory has a distinct interpretation. In a ring like an integral domain, an ideal is a set that absorbs multiplication by elements of the ring. When you hear of the ideal \(\langle a \rangle\), think of it as a collection of all possible products of \(a\) with every element in the ring.Let's break it down:

• The ideal \(\langle a \rangle\) contains all elements of the form \(ra\) where \(r\) is any element from the ring \(R\).
• Now, when two ideals \(\langle a \rangle\) and \(\langle b \rangle\) are equal, every element produced by multiplying \(a\) with elements of \(R\) can also be produced by multiplying \(b\) with elements of \(R\), and vice versa.

This equivalence forms a basis to conclude similarities or associations between \(a\) and \(b\), as the entire "reach" of these elements within the ring is identical.

Equivalence

In mathematical terms, equivalence often means there is a bidirectional relationship or mutual association between concepts. For this exercise, when associating elements in an integral domain, we use conditions of equivalence.Equivalence Structures:

• Starting with the equivalence relation \(a \sim b\), if \(a\) divides \(b\) and \(b\) divides \(a\), then we're looking at them as interchangeable through associates.
• Also, equivalence is seen in ideal terms: when \(\langle a \rangle = \langle b \rangle\), demonstrating how their generated 'power' in the ring is indistinguishable.

Understanding these equivalences proves that the relationships involved preserve the structure and properties of the integral domain, ensuring that either view, whether through divisibility or ideal equality, leads to the same conclusion.