Tag Archives: irreducibles

Commutative Algebra 15

Posted on April 2, 2020 by limsup

Unique Factorization Through this article and the next few ones, we will explore unique factorization in rings. The inspiration, of course, comes from ℤ. Here is an application of unique factorization. Warning: not all steps may make sense to 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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Topics in Commutative Rings: Unique Factorisation (1)

Posted on November 2, 2012 by limsup

Unique Factorisation: Basics Throughout this post, let R be an integral domain; recall that this means R is a commutative ring such that whenever ab=0, either a=0 or b=0. The simplest example of an integral domain is Z, the ring of integers. What’s of interest to … Continue reading →

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