Question

Show that if \(S\) is a subring of a ring \(R,\) then a set \(T \subseteq S\) is a subring of \(R\) if and only if \(T\) is a subring of \(S\).

Short answer

Question: Show that if S is a subring of a ring R, then a set T ⊆ S is a subring of R if and only if T is a subring of S. Answer: T is a subring of R if and only if T is a subring of S.

Step-by-step solution

Assume T is a subring of S

We need to show that T is also a subring of R. To prove this, we must show that T satisfies the subring test with respect to R. There are three conditions we need to verify: 1. T is non-empty. 2. If a and b are in T, then a - b is in T. 3. If a and b are in T, then ab is in T. Since T is a subring of S, it satisfies the subring test with respect to S. Therefore, T is non-empty and must contain at least one element. Then, let a and b be elements of T: 1. If T is non-empty, then there exists at least one element in T, which means there is at least one element in S that belongs to R. So, T is non-empty with respect to R. 2. Since a, b ∈ T ⊆ S, we know that a - b ∈ S. But S is a subring of R, so a - b ∈ R. Thus, a - b is in T with respect to R. 3. Similarly, since a, b ∈ T ⊆ S, we have ab ∈ S. Since S is a subring of R, ab ∈ R. Thus, ab is in T with respect to R. Therefore, T is a subring of R.

Assume T is a subring of R

Now we need to show that T is also a subring of S. To prove this, we must show that T satisfies the subring test with respect to S. We can use the same conditions as before: 1. T is non-empty. 2. If a and b are in T, then a - b is in T. 3. If a and b are in T, then ab is in T. Since T is a subring of R, it satisfies the subring test with respect to R. Therefore, T is non-empty and must contain at least one element. Then, let a and b be elements of T: 1. If T is non-empty with respect to R, then there exists at least one element in T, which also belongs to S as T ⊆ S. So, T is non-empty with respect to S. 2. Since a, b ∈ T ⊆ R, we know that a - b ∈ R. But T ⊆ S and S is a subring of R, so a - b ∈ S. Thus, a - b is in T with respect to S. 3. Similarly, since a, b ∈ T ⊆ R, we have ab ∈ R. Since T ⊆ S and S is a subring of R, ab ∈ S. Thus, ab is in T with respect to S. Therefore, T is a subring of S.

Conclusion

We have shown that if T is a subring of S, then T is also a subring of R, and if T is a subring of R, then T is also a subring of S. Thus, T is a subring of R if and only if T is a subring of S.

Key concepts

Ring Theory

Ring theory is a fundamental area of abstract algebra that deals with algebraic structures known as 'rings'. A ring is a set equipped with two binary operations—usually referred to as addition and multiplication—that generalize the arithmetic properties of the integers. In ring theory, we typically require that addition within a ring forms an abelian group, meaning it is commutative, associative, has an identity element (zero), and each element has an inverse.

Further, multiplication in a ring must be associative, and it should distribute over addition, following the distributive law. However, rings do not necessarily have a multiplicative identity (making them a more general concept), and elements may not have multiplicative inverses. The study of ring theory not only includes the analysis of individual rings but also the relationships between rings, such as homomorphisms and subrings, which are the main focus of our exercise.

Number Theory

Number theory is a branch of pure mathematics that revolves around the study of integers and integer-valued functions. It has a rich history with problems dating back thousands of years and has been called the 'queen of mathematics' by Gauss. This field of study focuses on understanding the properties and behaviors of numbers, especially concerning divisibility, prime numbers, modular arithmetic, and the solutions to equations in integers.

While it might appear that number theory and ring theory are distinct, they are deeply connected. In fact, rings provide an essential framework for many areas of number theory. For example, sets of numbers like integers, rational numbers, and modular integers form rings that are central to number-theoretical studies. These structures facilitate the development of advanced number-theoretical concepts, such as congruences and residue classes, which are integral to solving many problems in the field.

Algebra

Algebra is a significant branch of mathematics that deals with symbols and the rules for manipulating these symbols; it is the unifying thread of almost all of mathematics. From solving elementary equations to developing complex abstract structures, algebra provides a language to represent problems and operations. Within algebra, we distinguish between elementary algebra, focusing on the general properties of operations and their application to solving equations, and abstract algebra, which introduces algebraic structures like groups, rings, and fields.

Abstract algebra takes the concepts from elementary algebra to a higher level. It abstracts the notion of arithmetic operations and provides a rigorous framework for studying more complex structures that arise in various mathematical areas, including number theory, geometry, and beyond. The exercise we are discussing entwines abstract algebra's facets, particularly the concept of subrings, which are akin to subsets but uphold algebraic operations within a larger ring.

Subring Test

The subring test is a criterion used in algebra to determine whether a subset of a ring constitutes a subring. A subring is a subset that retains the ring structure; that is, if the larger set is a ring, then the operations of addition and multiplication within the subring should also satisfy the ring properties. According to the subring test, a non-empty subset T of a ring R is a subring if it satisfies two primary conditions:

• For any elements a and b in T, the difference a - b is also in T.
• For any elements a and b in T, the product ab is also in T.

If these conditions are met, then T inherits the ring operations from R and is thus considered a subring. Our exercise applied the subring test to establish the relationship between subrings of a given ring, demonstrating that if a set is a subring of 'any' subring of a ring, then it is also a subring of the ring itself, and vice versa. This logical equivalence is significant as it reveals the innate structure-preserving nature of subrings within algebraic systems.