Question

Suppose that \(R\) is a non-trivial ring in which the cancellation law holds in general: for all \(a, b, c \in R,\) if \(a \neq 0\) and \(a b=a c,\) then \(b=c .\) Show that \(R\) is an integral domain.

Short answer

Question: Prove that a non-trivial ring R that obeys the cancellation law is also an integral domain. Answer: To prove that a non-trivial ring R that obeys the cancellation law is an integral domain, we have shown that if a and b are non-zero elements of the ring such that ab = 0, then either a = 0 or b = 0, as required for an integral domain. The contradiction between our assumption and the cancellation law allows us to conclude that the product of non-zero elements in R cannot be 0, hence R has no zero divisors and is an integral domain.

Step-by-step solution

Recap of the Cancellation Law and Integral Domain

The cancellation law states that for all \(a, b, c\in R\), if \(a \neq 0\) and \(ab = ac\), then \(b = c\). To show that the ring \(R\) is an integral domain, we must prove that for all \(a, b \in R\), if \(a b=0\), then either \(a=0\) or \(b=0\).

Assume \(a\) and \(b\) are Non-Zero Elements of \(R\) such that \(ab=0\)

Let's assume \(a\) and \(b\) are non-zero elements in \(R\) such that \(a b =0\). We need to show that this cannot happen if the cancellation law holds in the ring \(R\). In other words, the cancellation law will lead to a contradiction if it holds in the ring \(R\).

Use the Cancellation Law

Using the cancellation law, since \(a \neq 0\) and \(ab = a(0)\), we would have \(b = 0\). However, in our assumption, we claimed that \(a\) and \(b\) are non-zero elements of \(R\) such that \(ab = 0\). This contradicts the cancellation law, implying that our assumption is incorrect.

Conclude the Proof

As a result of the contradiction found in step 3, we can conclude that if \(a\) and \(b\) are non-zero elements of the ring such that \(ab = 0\), then \(a\) or \(b\) must be \(0\) (i.e., the product of non-zero elements in \(R\) cannot be \(0\)). Hence, \(R\) is an integral domain since it has no zero divisors.

Key concepts

Cancellation Law in Ring Theory

The cancellation law is a fundamental concept in ring theory that establishes a property of certain algebraic structures known as rings. In essence, the cancellation law allows one to deduce equality of elements when multiplying by a common non-zero element.

The law states that for any elements a, b, and c in a ring R, if a is non-zero and the product of a and b equals the product of a and c (that is, if ab = ac), then b must equal c. This rule resembles the process of 'canceling out' the common factor on both sides of a fraction in basic arithmetic.

However, the cancellation law doesn't hold in all rings. It is valid only in rings without zero divisors. A zero divisor is an element that is non-zero itself but when multiplied with another non-zero element, yields zero. For example, if we have a ring with a zero divisor d, and two non-zero elements x and y such that dx = 0 and dy = 0, the cancellation law would incorrectly suggest that x equals y, which is not necessarily true. Thus, when the cancellation law is universally applicable in a ring, it implies the absence of zero divisors, characterizing an integral domain.

Ring Theory

Ring theory is the study of rings—a type of mathematical structure that generalizes fields, where addition, subtraction, and multiplication are defined and behave in a familiar way. However, unlike fields, rings are not required to have a multiplicative inverse for every non-zero element.

A ring is, more precisely, a set equipped with two binary operations (typically called addition and multiplication) that satisfy the properties of being an abelian group under addition and a monoid under multiplication, where multiplication is also distributive over addition.

An example of a ring is the set of integers, \( \mathbb{Z} \), which includes the familiar operations of addition and multiplication. In teaching these concepts, it's vital to stress that while every field is a ring, not every ring is a field. Integral domains form a special class of rings that are 'almost' fields—they lack zero divisors, and that exclusivity brings us closer to field-like behavior.

Zero Divisors

Zero divisors are particular elements in a ring that challenge the intuitiveness of multiplication's non-null outcomes. Formally, an element 'a' in a ring R is called a zero divisor if there exists a non-zero element 'b' such that the product of a and b is zero (i.e., ab = 0) but neither a nor b is zero.

Why are zero divisors so important? They are critical roadblocks to certain properties like the cancellation law. In a ring that has zero divisors, you can't safely cancel out terms like in integral domains or fields. This means that solving equations within such rings can be more complex and counterintuitive, as the 'zero product property' does not hold.

To relate this to real life, imagine if putting two non-empty containers together (multiplying) sometimes resulted in both containers appearing empty—the seemingly illogical outcome is analogous to what zero divisors do in the context of ring theory. This peculiarity is vital to understand for students delving into more advanced topics in algebra.