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8 Cards in this Set
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Definition: Ring |
Let R be a non-empty set along with two closed binary operations + and • (R, +, •) is called a ring if ∀a,b,c∈R
1) + is -Commutitive -Associative -Has an identity element -All elements have inverses 2) • is -Associative 3) • distributes over + -meaning: -a•(b+c) = (a•b)+(a•c), and -(a+b)•c = (a•c)+(b•c) Ex. (Z,+,•) is a ring, (R,+,•) is a ring |
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Definition: Unity |
Suppose (R, +, •) is a ring If ∃a∈R, such that ∀x∈R, a • x = x, we call 'a' a unity. And we call R a ring with unity. |
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Definition: Unit |
Suppose (R,+,•) is a ring with unity. If a∈R had a multiplicative inverse (∃b∈R, a•b = b•a = [unity]) we call 'a' a unit Ex. (Z,+,•) has units +-1 (R,+,•) everything but 0 is a unit |
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Definition: Commutitive Ring |
If (R,+,•) is a ring where ∀a,b∈R [a • b = b •a] Note: we call the z for which a + z = a, a zero element |
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Definition: Field |
Let (R,+,•) be a commutitive ring with unity. If every non-zero element is a unit we call the ring a field Ex. (Z,+,•) is not a field (Q,+,•) and (R,+,•) are fields |
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Definition: Proper Divisor |
a and b are proper divisors of zero if ab = 0 but a,b =/= 0 |
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Definition: Integral Domain |
A commutitive ring R with unity is called an integral domain if it has no proper divisor of zero Ex. (Z,+,•), (Q,+,•),(R,+,•) are integral domains Look at notes for one that isn't |
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Theorms: In any ring (R,+,•), ∀a,b,c∈R |
2) The additive inverse of a is unique (called -a) 3) a+b = a+c => b=c b+a = c+a => b=c 4) If 0 is the zero element, a0 = o 5) -(-a) = a 6) a(-b) = (-a)b = -(ab) 7) (-a)(-b) = ab 8) If R had a unity a) it is unique (call it u) b) the multiplicative inverse of a is unique if it exists (call it a⁻¹) 9) R is an integral domain iff ∀a,b,c, a=/=0, ab=ac => b=c 10) If (F,+,•) is a field then (F,+,•) is an integral domain 11) If R is not finite and an integral domain, then it is a field |
