Flashcards
MATH 228 Final - Definitions
Cards in this deck (7)
- 1Let R be an integral domain. Define an irreducible element in R.An element a in R is irreducible if:
i) a is not zero or a unit
ii) the only units dividing a are the units u and the elements ua. - 2Let R be an integral domain. Define a prime element in R.An element a in R is prime if
i) a is not zero or a unit
ii) given any equation a|bc, it follows that a|b or a|c (or both). - 3Let R and S be commutative rings. Define a ring homomorphism from R to S.A function f: R -> S is a ring homomorphism if
i) f(x+y) = f(x)+f(y) and f(xy) = f(x)f(y) for all classes x, y in R
ii) f(1R) = 1S.
Also note that if f is a homomorphism, then f(0R) = 0S follows from the ring axioms. - 4Give the definition of a field.A field is a commutative ring F with two properties:
i) 1F != 0F
ii) for every non-zero element a in F, there exists x in F such that ax = 1. (For each non-zero a, this x is written a^(-1), so we have (a)(a^-1) = 1 for every non-zero a in F. - 5Let R be an integral domain. Define a greatest common divisor in R.Given a set of elements a_1, ..., a_n in R, a greatest common divisor of the a_i is any element d with the following two properties:
i) d divides every a_i
ii) if an element b in R divides every a_i, then b|d. - 6Define an equivalence relation.Let X be a non-empty set. An equivalence relation = on X is a relation which has the three properties:
i) a = a for all a in X (reflexivity)
ii) if a = b, then b = a (symmetry)
iii) if a = b and b = c, then a = c (transitivity) - 7Give the definition of an ideal.Let R be a commutative ring. A subset I is called an ideal if
i) 0R is in I
ii) if a, b are in I, then a+b is in I
iii) if a is in I and r is in R, then ar is in I
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- Mathematics Flashcards on MATH 228 Final - Definitions, created by Chuck Hanson on 16/04/2018.
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