Question

Let \(R\) be a ring, and let \(a, b \in R\) such that \(a b \neq 0 .\) Show that \(a b\) is a zero divisor if and only if \(a\) is a zero divisor or \(b\) is a zero divisor.

Short answer

Based on the step-by-step solution provided, the short answer is: To show that the product of two elements, \(ab\), in a ring \(R\) is a zero divisor if and only if either \(a\) or \(b\) is a zero divisor, we prove two claims: 1. If either \(a\) or \(b\) is a zero divisor, then \(ab\) is a zero divisor. Assume \(a\) is a zero divisor. Then, there exists a nonzero element \(r\in R\) such that \(ar=0\). We then have \((ab)r = a(br) = a(0) = 0\). Since \(ab\) multiplied by the nonzero element \(r\) is equal to 0, \(ab\) is a zero divisor. 2. If \(ab\) is a zero divisor, then either \(a\) or \(b\) must be a zero divisor. We prove this claim by proving its contrapositive: If neither \(a\) nor \(b\) are zero divisors, then \(ab\) is not a zero divisor. Assume \(a\) and \(b\) are not zero divisors. If there exists a nonzero element \(r'\) such that \((ab)r' = 0\), then multiplying both sides by \(a^{-1}\) and \(b^{-1}\) gives: \(A((ab)r')B = a(0) = 0\). Since \(a\) is not a zero divisor, \(r'=0\), which contradicts the assumption that \(ab\) is a zero divisor. Therefore, if neither \(a\) nor \(b\) are zero divisors, then \(ab\) is not a zero divisor. In conclusion, \(ab\) is a zero divisor if and only if either \(a\) or \(b\) is a zero divisor.

Step-by-step solution

Proving that if either a or b is a zero divisor, then ab is also a zero divisor.

Assume that either \(a\) or \(b\) is a zero divisor. Without loss of generality, let's assume that \(a\) is a zero divisor. This means there exists a nonzero element \(r\in R\) such that \(ar = 0\). Now, we want to show that \(ab\) is a zero divisor. So let's multiply \(abr\) by some nonzero element in \(R\). Since we already know that \(ar=0\), we have: \((ab)r=a(br)=a(0)=0\). Since \(ab\) multiplied by the nonzero element \(r\) is equal to 0, \(ab\) is a zero divisor.

Proving that if ab is a zero divisor, then either a or b must be a zero divisor.

Now, let's assume that \(ab\) is a zero divisor. So, there exists a nonzero element \(r'\in R\) such that \((ab)r' = 0\). To prove our main claim, we will prove the contrapositive of our statement, i.e., if neither \(a\) nor \(b\) are zero divisors, then \(ab\) is not a zero divisor. Assume that \(a\) and \(b\) are not zero divisors. We want to show that \(ab\) is not a zero divisor either. Now, consider the equation \((ab)r'=0\). Let \(A=a^{-1}\) and \(B=b^{-1}\). Multiplying both sides by \(A\) and \(B\), we get: \(A(a(b(r')))B \Rightarrow A((ab)r')B \Rightarrow (A0)B \Rightarrow a(0)=0\). Now, since \(a\) is not a zero divisor, \(r'=0\) (if \(r'\neq 0\), \(a\) would have been a zero divisor with the multiplier \(b(r'))\). This contradicts the assumption that \(ab\) is a zero divisor, because in this case there is no nonzero element \(r'\) such that \((ab)r'=0\). Thus, we have proven that if neither \(a\) nor \(b\) are zero divisors, then \(ab\) is not a zero divisor. This is equivalent to our original statement, which says if \(ab\) is a zero divisor, then \(a\) or \(b\) must be zero divisors.

Key concepts

Ring Theory

Ring theory is a fundamental area within abstract algebra that studies algebraic structures known as rings. A ring is a set equipped with two binary operations that we often call addition and multiplication, following the behavior of integers. These operations must satisfy certain conditions: addition must form an abelian group, and multiplication must be associative. Furthermore, multiplication must distribute over addition.

When discussing ring theory, it's essential to comprehend that the elements of a ring can interact in complex ways, leading to interesting properties and results. As in the original exercise, where we deal with the product of two elements within a ring, ring theory helps in understanding how these elements can give rise to a zero divisor, which is a key concept in exploring the structure and characteristics of rings.

Zero Divisor Properties

In the context of ring theory, a zero divisor is an element in a ring that can produce a zero product when multiplied by a nonzero element. The presence of zero divisors has significant implications for the structure of the ring.

Understanding zero divisors is crucial because they illuminate how multiplication within a ring can deviate from the behavior of familiar number systems like the integers. Some key properties of zero divisors include:

• Zero divisors are not invertible; if an element has a multiplicative inverse, it can't be a zero divisor.
• Elements that are zero divisors cannot be units or elements with multiplicative inverses within the ring.
• If a ring has no zero divisors, it is referred to as an integral domain, a special kind of ring.

In the exercise example, the concept of zero divisor is essential for the logical argumentation involved in showing the relationship between the product of elements and the zero divisor property.

Abstract Algebra

Abstract algebra is a broad area of mathematics, dealing with algebraic structures such as groups, rings, fields, and modules. It aims to generalize and abstract the concepts of algebra beyond familiar numbers, focusing on the deeper properties of operations and structures.

The concepts in abstract algebra, such as zero divisors, arise naturally as we explore more general algebraic systems that do not behave like the number systems we are accustomed to. For instance, in the multiplicative system of a ring, encountering an element that nullifies another nonzero element (zero divisor) challenges our basic understanding of multiplication as it operates differently from the multiplication of real numbers.

Applying abstract algebraic thinking to solve problems, as in the given exercise, strengthens the understanding of these advanced concepts. It underscores the importance and interconnectedness of properties like zero divisors in determining the nature of the algebraic structures being studied.