Polynomial Rings: PID vs UFD Properties

This question asks us to identify the statement that is NOT true regarding polynomial rings over the integers (\(\mathbb{Z}\)) and the rational numbers (\(\mathbb{Q}\)), specifically concerning the properties of being a Principal Ideal Domain (PID) and a Unique Factorization Domain (UFD).

Understanding PID and UFD

Let's first understand what PID and UFD mean for a ring R:

• Principal Ideal Domain (PID): An integral domain R is a PID if every ideal in R is principal, meaning it can be generated by a single element. Formally, for every ideal I in R, there exists an element a ∈ R such that I = <a> = {ra | r ∈ R}.
• Unique Factorization Domain (UFD): An integral domain R is a UFD if every non-zero, non-unit element in R can be written as a product of irreducible elements, and this factorization is unique up to associates and the order of the factors.

A key relationship between these concepts is that every PID is also a UFD.

Analyzing Polynomial Rings

We need to consider the properties of polynomial rings \(R[x]\) where R is an integral domain.

Polynomial Ring &(\mathbb{Q}\)[x]

• The coefficient ring here is \(\mathbb{Q}\), the field of rational numbers.
• A well-known theorem states that for a field F, the polynomial ring \(F[x]\) is always a Principal Ideal Domain (PID). Since \(\mathbb{Q}\) is a field, \(\mathbb{Q}[x]\) is a PID.
• As every PID is also a UFD, it follows that \(\mathbb{Q}[x]\) is also a Unique Factorization Domain (UFD).

Based on this, the statements about \(\mathbb{Q}[x]\) are:

• Statement 2: The polynomial ring \(\mathbb{Q}[x]\) is a Principal Ideal Domain (PID). This is TRUE.
• Statement 4: The polynomial ring \(\mathbb{Q}[x]\) is a Unique Factorization Domain (UFD). This is TRUE (since it's a PID).

Polynomial Ring &(\mathbb{Z}\)[x]

• The coefficient ring here is \(\mathbb{Z}\), the ring of integers. \(\mathbb{Z}\) is an integral domain, but it is NOT a field (since not every non-zero element has a multiplicative inverse, e.g., 2).
• A polynomial ring \(R[x]\) over an integral domain R is a PID if and only if R is a field. Since \(\mathbb{Z}\) is not a field, the polynomial ring \(\mathbb{Z}[x]\) is NOT a PID.
• To see why \(\mathbb{Z}[x]\) is not a PID, consider the ideal I = <2, x> generated by the elements 2 and x in \(\mathbb{Z}[x]\). This ideal consists of all polynomials of the form \(2 \cdot p(x) + x \cdot q(x)\) where \(p(x), q(x) \in \mathbb{Z}[x]\). If this ideal were principal, it would be generated by a single polynomial \(f(x)\), i.e., I = <f(x)>. Since 2 is in I, \(f(x)\) must divide 2 in \(\mathbb{Z}[x]\). This means \(f(x)\) must be a constant, either 1, -1, 2, or -2. If \(f(x)\) were 1 or -1, then I would be <1> = \(\mathbb{Z}[x]\). But I does not contain 1 (or any non-zero constant other than possibly 2 or -2 multiples) because any element in I evaluated at x=0 gives an even integer. So 1 is not in I. Thus \(f(x)\) must be 2 or -2. Let's assume \(f(x) = 2\). Then I = <2>. This would mean x is a multiple of 2 in \(\mathbb{Z}[x]\), i.e., \(x = 2 \cdot g(x)\) for some \(g(x) \in \mathbb{Z}[x]\). This is impossible because the coefficients of \(g(x)\) are integers, and \(2 \cdot g(x)\) would have only even coefficients, while x has coefficient 1. Thus, the ideal <2, x> cannot be generated by a single element, and \(\mathbb{Z}[x]\) is not a PID.
• However, a theorem by Gauss states that if R is a UFD, then the polynomial ring \(R[x]\) is also a UFD. Since \(\mathbb{Z}\) is a PID (and thus a UFD), \(\mathbb{Z}[x]\) is a Unique Factorization Domain (UFD).

Based on this, the statements about \(\mathbb{Z}[x]\) are:

• Statement 1: The polynomial ring \(\mathbb{Z}[x]\) is a Principal Ideal Domain (PID). This is FALSE.
• Statement 3: The polynomial ring \(\mathbb{Z}[x]\) is a Unique Factorization Domain (UFD). This is TRUE (by Gauss's Lemma, since \(\mathbb{Z}\) is a UFD).

Conclusion

We are looking for the statement that is NOT true. Based on our analysis:

The statement that is NOT true is "The polynomial ring \(\mathbb{Z}[x]\) is a Principal Ideal Domain (PID)".