Tag Archives: principal ideal domains

Commutative Algebra 17

Posted on April 4, 2020 by limsup

Field of Fractions Throughout this article, A denotes an integral domain (which may not be a UFD). Definition. The field of fractions of A is an embedding of A into a field K, such that every element of K can be … Continue reading →

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Commutative Algebra 16

Posted on April 3, 2020 by limsup

Gcd and Lcm We assume A is an integral domain throughout this article. If A is a UFD, we can define the gcd (greatest common divisor) and lcm (lowest common multiple) of two elements as follows. For , we can write the … Continue reading →

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Topics in Commutative Rings: Unique Factorisation (3)

Posted on November 7, 2012 by limsup

Example 1: The Gaussian Integers Z[i] Let’s pick the norm function N : Z[i]-{0} → N where N(a+bi) = (a+bi)(a–bi) = a2+b2. We know that N is a multiplicative function, i.e. N(r)N(s) = N(rs). Instead of checking this by brute force, we write N(x) = x·xc, where (a+bi)c = a-bi is the conjugate of a+bi. It’s easy to … Continue reading →

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Topics in Commutative Rings: Unique Factorisation (2)

Posted on November 4, 2012 by limsup

In the previous article, we imposed certain finiteness conditions on the ring (specifically a.c.c. on principal ideals: that every increasing sequence of principal ideals is eventually constant), then proved that unique factorisation holds if and only if all irreducible elements … Continue reading →

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