Problem 6 - A Book of Abstract Algebra

Question

Let \(A\) be any ring. Prove that if the additive group of \(A\) is cyclic, then \(A\) is a commutative ring

Step-by-Step Solution

Answer

Since the multiplicative operation ensures \(xy = yx\) for arbitrary elements, \(A\) is commutative.

Step 1: Understanding the Problem

We are given a ring \(A\) whose additive group is cyclic and asked to prove that this implies \(A\) is a commutative ring. A cyclic group is one that can be generated by a single element.

Step 2: Define a Cyclic Additive Group

Let the additive group of \(A\) be generated by an element \(g\), i.e., every element \(a\) in \(A\) can be expressed as \(a = ng\) for some integer \(n\).

Step 3: Select Arbitrary Elements

To prove commutativity, take two arbitrary elements \(x"), y \) in \(A\). Since the additive group is cyclic, there exist integers \(m\) and \(n\) such that \(x = mg\) and \(y = ng\).

Step 4: Use Ring Multiplication

Consider the product \(xy\). We have \(xy = (mg)(ng) = m(ng^2)\). Since the result must reside in the sub-group generated by \(g\), write \(g^2 = kg\) for some integer \(k\). Thus, \(xy = m(nk)g\).

Step 5: Verify Commutativity

Now consider \(yx = (ng)(mg) = n(mk)g\). To prove commutativity, we require \(xy = yx\), which translates to \(m(nk)g = n(mk)g\). This holds because multiplication of integers is commutative.

Step 6: Conclude with Commutativity

Since arbitrary elements \(x\), \(y\) satisfy \(xy = yx\), the ring \(A\) is commutative under multiplication.

Key Concepts

Cyclic Group

Cyclic groups are an important concept in the study of abstract algebra and group theory. A group is termed cyclic if there is an element within the group such that every other element of the group can be expressed as a power of this element. In other words, a cyclic group is generated by a single element.
The significance of cyclic groups lies in their simplicity and structure. They can be infinite, like the group of integers under addition, or finite, like the integers modulo a number under addition.
For example, if you consider a cyclic group generated by an element \( g \), every element \( a \) in the group can be written as \( a = ng \) where \( n \) is an integer. In our context, this property dramatically simplifies the structure of the additive group of a ring, making it predictable and manageable. This regularity is crucial in proving complex properties, such as commutativity in rings, connecting back to our problem.

Ring Theory

Ring Theory serves as the mathematical stage for studying rings, which are algebraic structures consisting of a set equipped with two binary operations. These operations are typically addition and multiplication.
For a set to qualify as a ring, it must satisfy certain rules or axioms, including associative properties for addition and multiplication, distributivity of multiplication over addition, and, notably, a unity element for addition.
Rings can be commutative or non-commutative. A commutative ring is one where the multiplication operation is commutative, that is, for any two elements \( a \) and \( b \), \( ab = ba \). This property is crucial in many areas of algebra and is precisely what we aim to establish in the exercise from a cyclic additive group assumption.

Proof Writing

Proof writing is an essential skill in mathematics, allowing us to establish the truth of statements in a logical and structured manner. Writing proofs involves providing a series of logical steps that connect what's given to what's to be shown.
In our exercise, we begin by understanding the problem and identifying what needs to be proven: if a ring's additive group is cyclic, then the ring itself must be commutative.
The proof consists of:

• Clarifying definitions and assumptions, like the nature of the cyclic group.
• Using arbitrary but specific elements to demonstrate the general case.
• Pursuing a logical sequence of deductions involving ring properties.
• Finally, making a conclusion that ties back to the original statement—proving commutativity under the given conditions.

This approach not only tests understanding but also solidifies concepts by linking ideas in a coherent and logical way.

Abstract Algebra

Abstract algebra is a broad area of mathematics that deals with algebraic structures such as groups, rings, and fields, providing a unifying framework to understand these and other related concepts.
At its core, abstract algebra provides the tools and language needed to explore and reason about the algebraic properties (e.g., operations and relations) of mathematical structures.
In the context of our exercise, abstract algebra gives us the foundational terminology and concepts: rings, groups, and their properties. It allows us to handle operations abstractly rather than with specific numbers or functions.
Through this lens, we explore the simplicity and consequences of having a cyclic additive group, leveraging these abstract concepts to deduce properties of the ring itself. Understanding such intricate relationships is a key strength of abstract algebra, contributing to deeper insights across mathematics.