91Ó°ÊÓ

A subdomain is a subset of a larger domain that is structured in such a way that it retains the operational properties of the larger set, like addition and multiplication. In the context of integral domains, if you have a set, like our set \( S = \{n \cdot 1 \mid n \in \mathbb{Z}\} \), to qualify as a subdomain of \( D \), it must satisfy certain properties. The set \( S \) is formed by multiplying each integer \( n \) with the multiplicative identity \( 1 \) of \( D \). Thus, every element in \( S \) is essentially \( n \cdot 1 \), which is definitely included in \( D \) because domains close a group structure under their defined operations.
A subdomain must inherit properties like closure under addition and multiplication from its parent domain. Furthermore, it must include specific elements like the additive and multiplicative identities to ensure that it's a complete, intact section of the domain it resides within. This guarantees that operations within \( S \) behave identically to those in \( D \).

The concept of additive identity is crucial in mathematical sets. The additive identity is an element in a set that, when added to any element of that set, does not change that element's value. For most numerical contexts, including integral domains like our set \( S \), the additive identity is the number \( 0 \).
In our set \( S = \{n \cdot 1 \mid n \in \mathbb{Z}\} \), the element \( 0 \) must be a part of \( S \) for it to be considered a subdomain. We verify this by noting that \( 0 \cdot 1 = 0 \), which fits the pattern of elements in \( S \). This means that adding \( 0 \) to any element of \( S \) results in the same element, thereby fulfilling the condition for the additive identity.

• The number \( 0 \) is such that for any \( a \) in \( S \), \( a + 0 = a \).
• Preserving this property is essential for \( S \) to act as a subdomain within \( D \).

The multiplicative identity of a set is an element that, when multiplied with any other element of the set, returns the original element. In an integral domain, this identity is often represented by \( 1 \). It's what allows multiplication within the domain to behave predictably and coherently.
In our context, \( S \) must include this identity element. Notice how every element of \( S \) takes the form \( n \cdot 1 \), meaning \( 1 \), the multiplicative identity of \( D \), is inherently part of \( S \). Thus, for any element \( a = n \cdot 1 \) in \( S \), multiplying by \( 1 \) leaves it unchanged, \( a \cdot 1 = a \), confirming that \( 1 \) acts as the multiplicative identity within this context.

• Inclusion of \( 1 \) ensures that \( S \) supports multiplicative operations appropriately.
• For \( S \) to be a subdomain, it should indeed contain the multiplicative identity of \( D \).

Closure under operations ensures that performing internal operations like addition and multiplication on elements of a set results in elements that are still within the set. This is critical for any subset to be considered a subdomain, particularly in algebraic structures like rings or integral domains.
For our specific set \( S \), we need to show closure under both addition and multiplication.

• **Closure under addition:** Taking two elements \( a = m \cdot 1 \) and \( b = n \cdot 1 \), their sum \( a + b = m \cdot 1 + n \cdot 1 = (m+n) \cdot 1 \) remains in \( S \).
• **Closure under multiplication:** Similarly, multiplying \( a = m \cdot 1 \) and \( b = n \cdot 1 \) gives \( a \cdot b = (m \cdot n) \cdot 1 \), and since \( m \cdot n \in \mathbb{Z} \), it remains in \( S \).

These proofs establish that \( S \) maintains its integrity under these two core operations, thereby operating as a legitimate subdomain of \( D \). The operations do not lead outside of \( S \), which is a fundamental requirement for closure.