Tag Archives: UFDs

Commutative Algebra 53

Graded Rings Definition. A grading on a ring A is a collection of additive subgroups such that as abelian groups, and for any , i.e.. A graded ring is a ring A with a specified grading. Note The notation means every … Continue reading

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Commutative Algebra 39

Integrality Throughout this article, A is a subring of B; we will also call B a ring extension of A. Definition. An element is said to be integral over A if we can find (where ) such that in B. For example, is integral over since … Continue reading

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Commutative Algebra 17

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 15

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)

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)

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)

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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