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Visual Group Theory: Lecture 7.4: Divisibility and factorization
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Visual Group Theory: Lecture 7.4: Divisibility and factorization
The ring of integers have a number of properties that we take for granted: every number can be factored uniquely into primes, and all pairs of numbers have a unique gcd and lcm. In this lecture, we investigate when this happens in other integral domains. We define what it means for elements to be prime and to be irreducible. Prime implies irreducible, except the converse fails unless we are in a special ring called a "principal ideal domain" (PID). There are rings for which every ideal is generated by a single element, and they have other nice properties. For example, every pair of elements in a PID has a unique gcd and lcm. A larger class of rings, for which some of these properties still old are the unique factorization domains (UFD). We conclude with a summary of the types of rings we've seen so far: fields, PIDs, UFD, integral domains, and commutative rings, and how they fit together.
The ring of integers have a number of properties that we take for granted: every number can be factored uniquely into primes, and all pairs of numbers have a unique gcd and lcm. In this lecture, we investigate when this happens in other integral domains. We define what it means for elements to be prime and to be irreducible. Prime implies irreducible, except the converse fails unless we are in a special ring called a "principal ideal domain" (PID). There are rings for which every ideal is generated by a single element, and they have other nice properties. For example, every pair of elements in a PID has a unique gcd and lcm. A larger class of rings, for which some of these properties still old are the unique factorization domains (UFD). We conclude with a summary of the types of rings we've seen so far: fields, PIDs, UFD, integral domains, and commutative rings, and how they fit together.
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