Karteikarten
MATH 228 Midterm 1 Content
Karten in diesem Stapel (9)
- 1Define an integral domain.An integral domain is a ring in which, for any a, b != 0, then ab also != 0.
- 2Define a unit.A unit is an element u for which there exists another element u^-1 such that (u)(u^-1) = 1. Worth noting is that u | 1.
- 3Define an irreducible element.In an integral domain, an element a is irreducible if a is not zero or a unit, and the only elements dividing a are u or ua.
(In notation: u|a or ua|a
=> a = ux or a = uax for some x) - 4Define a prime element.In an integral domain, an element p is prime if it is not zero or a unit, and if p|ab, then p|a or p|b (or both) for some a, b.
- 5Define a Unique Factorization Domain (UFD).A UFD is an integral domain in which:
- If an element a is not zero or a unit, it is a product of irreducibles
- For any two factorizations a = p1...pn = q1...qm, then m = n (i.e. there are the same number of irreducible factors) and pi = qi except for multiplication by various units. - 6Define a greatest common divisor.Given an integral domain and a set of elements a1,...,an , a greatest common divisor is an element d with the following properties:
- d divides ai for every ai in the set
- If an element b divides every ai in the set, then d | b. - 7Explain how you would prove that in an integral domain, all primes are irreducible.Let a be a prime element. Consider the factorization a = bc.
So bc = a(1) => a | bc (i.e. a|a).
=> Because a is prime, a|b or a|c.
Suppose a|b, so b = ax. Then a = bc => a = axc. We can cancel a:
=> 1 = xc => c is a unit.
Therefore a is irreducible. - 8How would you prove that the product of two units in a ring is again a unit?Let u, v be units of R, and let k = (u^-1), x = (v^-1).
=> uk = 1, vx = 1
=> (uk)(vx) = 1 => (uv)(kx) = 1
=> uv | 1
Thus uv is a unit. - 9—
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- Mathematics Karteikarten am MATH 228 Midterm 1 Content, erstellt von Chuck Hanson am 01/02/2018.
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