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Tag Archives: irreducibles
Topics in Commutative Rings: Unique Factorisation (3)
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 = abi is the conjugate of a+bi. It’s easy to … Continue reading
Topics in Commutative Rings: Unique Factorisation (2)
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
Posted in Notes
Tagged commutative rings, euclidean domains, irreducibles, prime ideals, primes, principal ideal domains, ring theory, rings, UFDs, unique factorisation
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Topics in Commutative Rings: Unique Factorisation (1)
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
Posted in Notes
Tagged commutative rings, irreducibles, prime ideals, primes, ring theory, rings, UFDs, unique factorisation
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