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Chapter 18: Problem 49

a. Show that an intenection of subrings of a ring \(R\) is again a subring of \(R\). b. Show that an intersection of subfields of a field \(F\) is again a subfield of \(F\).

Short Answer

Expert verified
Intersections of subrings and subfields are subrings and subfields, respectively.

Step by step solution

01

Define Subring

Recall that a subset \( S \) of a ring \( R \) is a subring if it is closed under addition, subtraction, and multiplication, and contains the multiplicative identity of \( R \).
02

Consider Intersection of Subrings

Let \( \{ S_i \}_{i \in I} \) be a collection of subrings of \( R \). Define \( S = \bigcap_{i \in I} S_i \). Show that \( S \) satisfies the subring properties.
03

Check Closure under Addition

For any \( a, b \in S \), since \( a,b \) are in each \( S_i \), their sum \( a+b \) must also be in each \( S_i \). Thus, \( a+b \in S \).
04

Check Closure under Subtraction

Since \( a, b \in S \) are in each \( S_i \), their difference \( a-b \) must also be in each \( S_i \), thus \( a-b \in S \).
05

Check Closure under Multiplication

For \( a, b \in S \), they are in each \( S_i \), so \( a \cdot b \) is in each \( S_i \). Thus, \( a \cdot b \in S \).
06

Check Identity Element

Since each \( S_i \) is a subring, the multiplicative identity \( 1 \) is in each \( S_i \), so \( 1 \in S \).
07

Define Subfield

Recall a subset \( T \) of a field \( F \) is a subfield if it is closed under addition, subtraction, multiplication, and division (except by zero), and contains \( 1 \) and \( 0 \).
08

Consider Intersection of Subfields

Let \( \{ T_j \}_{j \in J} \) be a collection of subfields of \( F \). Define \( T = \bigcap_{j \in J} T_j \). Show \( T \) satisfies the subfield properties.
09

Check Closure under Division

For any non-zero \( a \in T \), since \( a \) is in each \( T_j \), its inverse \( a^{-1} \) is also in each \( T_j \). Thus, \( a^{-1} \in T \).
10

Verify Other Field Properties

By previous steps verifying subring properties, and closure under division, \( T \) satisfies all field properties as does each \( T_j \).
11

Conclusion: Verify Both Statements

The intersection of subrings of a ring \( R \) is a subring, and the intersection of subfields of a field \( F \) is a subfield.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ring Theory
Ring theory is a fascinating area of mathematics focusing on the study of rings. A ring is a set equipped with two operations: addition and multiplication, that generalize the arithmetic of integers. Rings must satisfy several specific properties:
  • Closure under addition and multiplication: The sum and product of any two elements in the ring are also in the ring.
  • Associativity: Both addition and multiplication operations are associative.
  • An additive identity: There is an element, typically denoted as 0, which when added to any element in the ring returns the same element.
  • Additive inverses: For every element in the ring, there is another element such that their sum is the additive identity (0).
  • Distributive property: Multiplication distributes over addition.
These properties create the framework under which rings operate, allowing for both familiar and novel mathematical exploration. Understanding the basics of ring theory provides a critical foundation for diving deeper into more specific concepts such as subrings and field theory.
Field Theory
Field theory is an extension of ring theory, where we specifically consider fields, which are particular types of rings. A field is a set equipped with two operations, addition and multiplication, similar to rings but with stricter rules. The key distinctions for a field include:
  • Presence of both additive and multiplicative identities: While rings require only an additive identity, fields require both 0 (additive identity) and 1 (multiplicative identity).
  • Existence of multiplicative inverses: Every non-zero element in a field has a multiplicative inverse, such that the product of an element and its inverse is the multiplicative identity (1).
  • Commutativity: In fields, both addition and multiplication are commutative, meaning the order of operands does not affect the result.
Fields form the backbone of several branches of mathematics, including algebra, number theory, and even areas like cryptography. This rigorous structure allows for a wide array of mathematical techniques and theories to be developed from field theory.
Subring
A subring is simply a subset of a ring that is itself a ring with the operations inherited from the larger ring. For a subset to be considered a subring, it must fulfill the following conditions:
  • Closure under addition and subtraction: If you take any two elements in the subring, their sum and difference must also be in the subring.
  • Closure under multiplication: The product of any two elements in the subring should also remain in the subring.
  • Contains the multiplicative identity (if specified in the ring's definition of a subring).
These properties ensure that the structure and behavior of the subring are consistent with the ring itself. One important aspect of subrings is that the intersection of any collection of subrings is also a subring. This is a powerful concept since it shows that subrings interact well within the algebraic framework and maintain the ring properties.
Subfield
A subfield extends the concept of a subring by inheriting properties from the field. This means it has to satisfy even stricter criteria:
  • Closure under addition, subtraction, multiplication, and division (except division by zero): Operations conducted within a subfield must stay within the subfield.
  • Contains both the additive identity (0) and multiplicative identity (1): These identities ensure the integrity of the field operations.
To form a subfield, every element must have both an additive inverse within the subfield and, importantly, every non-zero element must have a multiplicative inverse within the subfield. A fascinating property of subfields is that the intersection of subfields results in another subfield. This maintains the field properties and supports further mathematical investigations within the realm of field theory.

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