Domains
We will cover following topics
Integral Domains
A commutative ring \(D\), with unity and having no divisors of zero, is called an integral domain.
Unit, Associate, Divisor
Let \(D\) be an integral domain.
Unit
An element \(v\) of \(D\) having a multiplicative inverse in \(D\) is called a unit (regular element) of \(D\).
Associate
An element \(b\) of \(D\) is called an associate of \(a \in D\) if \(b = v \cdot a\), where \(v\) is some unit of \(D\).
Divisor
An element \(a\) of \(D\) is a divisor of \(b \in D\) provided there exists an element \(c\) of \(D\) such that \(b = a \cdot c\).
Note: Every non-zero element \(b\) of \(D\) has as divisors its associates in \(D\) and the units of \(D\). These divisors are called trivial (improper); all other divisors, if any, are called non-trivial (proper).
Principal Ideal Domains
A principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.
Euclidean Domains
An integral domain \(R\) is called Euclidean if there is a function \(d: R− \{ 0 \} \rightarrow N\) with the following two properties:
i) \(d(a) ≤ d(ab)\) for all nonzero \(a\) and \(b\) in \(R\),
ii) for all \(a\) and \(b\) in \(R\) with \(b \neq 0\), we can find \(q\) and \(r\) in \(R\) such that
\(a = bq + r\), \(r = 0\) or \(d(r) < d(b)\).
Unique Factorization Domains
Irreducible
An element \(a\) in an integral domain \(R\) is called irreducible if it is not zero or a unit, and if whenever \(a\) is written as the product of two elements of \(R\), one of these is a unit.
Prime
An element \(p\) of an integral domain \(R\) is called prime if \(p\) is not zero or a unit, and whenever \(p\) divides \(ab\) for elements \(a\), \(b\) of \(R\), either \(p\) divides \(a\) or \(p\) divides \(b\).
Note that if \(R\) is an integral domain, every prime element of \(R\) is irreducible.
Associates
Elements \(r\) and \(s\) are called associates of each other if \(s = ur\) for a unit \(u\) of \(R\). So \(a \in R\) is irreducible if it can only be factorized as the product of a unit and one of its own associates.
Unique Factorization Domain
An integral domain \(R\) is a unique factorization domain if the following conditions hold for each element \(a\) of \(R\) that is neither zero nor a unit:
i) \(a\) can be written as the product of a finite number of irreducible elements of \(R\).
ii) This can be done in an essentially unique way. If \(a = p_1p_2 \cdots p_r\) and \(a = q_1q_2 \cdots q_s\) are two expressions for \(a\) as a product of irreducible elements, then \(s = r\) and \(q_1, \cdots , q_s\) can be reordered so that for each \(i\), \(q_i\) is an associate of \(p_i\).
PYQs
Integral Domains
1) Let \(R\) be an integral domain with unit element. Show that any unit in \(R(x)\) is a unit in \(R\).
[2018, 10M]
2) Let \(C=\{f : I=[0,1] \rightarrow R / f\) is continuous \(\}\). Show \(C\) is a commutative ring with 1 under point wise addition and multiplication. Determine whether \(C\)s is an integral domain. Explain.
[2010, 15M]
3) How many elements does the quotient ring \(\dfrac{Z_{5}[X]}{X^{2}+1}\) have? Is it an integral domain? Justify your answers.
[2009, 15M]
4) Define irreducible element and prime element in an integral domain \(D\) with units. Prove that every prime element in \(D\) is irreducible and converse of this is not (in general) true.
[2004, 25M]
Unique Factorization Domains
1) Let \(J=\{a+i b / a, b \in Z\}\) be the ring of Gaussian integers (subring of \(C\)). Which of the following is \(J\) - Euclidean domain, principal ideal domain, and unique factorization domain? Justify your answer.
[2013, 15M]
2) Show that \(Z[X]\) is a unique factorization domain that is not a principal ideal domain (\(Z\) is the ring of integers). Is it possible to give an example of principal ideal domain that is not a unique factorization domain? (\(Z[X]\) is the ring of polynomials in the variable \(X\) with integer.
[2009, 15M]
3) Let \(R\) be a ring with unity. If the product of any two non-zero elements is non-zero. Then prove that \(a b=1 \Rightarrow b a=1\). Whether \(Z_{6}\) has the above property or not explain. Is \(Z_{6}\) an integral domain?
[2008, 15M]
4) Show that
\[Z\left[ \sqrt{2} \right] = \left\{ a+ \sqrt{2} b \vert a, b \in Z \right\}\]is a Euclidean domain.
[2006, 30M]
5) Prove that any polynomial ring \(F[x]\) over a field \(F\) is U.F.D.
[2005, 30M]
6) If \(R\) is a unique factorization domain \((U . F . D)\), then prove that \(R[x]\) is also U.F.D.
[2003, 10M]
7) Show that the ring
\[Z[i] = \left\{ a+bi \vert a \in Z, b \in Z, i = \sqrt{-1} \right\}\]of Gaussian integers is a Euclidean domain.
[2003, 12M]