term1 Definition1term2 Definition2term3 Definition3
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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
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
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.
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
(Z,+,•) has units +-1
(R,+,•) everything but 0 is a unit
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
Definition: Field
Let (R,+,•) be a commutitive ring with unity. If every non-zero element is a unit we call the ring a field
(Z,+,•) is not a field
(Q,+,•) and (R,+,•) are fields
Definition: Proper Divisor
a and b are proper divisors of zero if ab = 0 but a,b =/= 0
Definition: Integral Domain
A commutitive ring R with unity is called an integral domain if it has no proper divisor of zero
(Z,+,•), (Q,+,•),(R,+,•) are integral domains
Look at notes for one that isn't
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
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