Groups, Rings, & Fields
Ring with Unity
A ring that has a multiplicative identity
(2Z, +, *)
commutative ring
Ordered Field
A field for which there is an order < on the elements. It has two basic properties: 1. If a < b and b < c then a < c 2. For any elements a and b of an ordered field, exactly one of the following is true: a < b, a = b, or b < a
Abelian Group
A group that also satisfies the commutative property under the binary operation.
Group
A non-empty set with a binary operation that satisfies the following properties: 1. The set is closed under the operation 2. The set is associative under the operation 3. There is an identity element, e, of X 4. There is an inverse element for every element that when applied will return the identity
Ring
A non-empty set with addition and multiplication (may look different than the add. and mult. we know) that satisfies the following properties: 1. The set under addition is an abelian group 2. The set under multiplication is associative 3. The set under addition and multiplication is distributive
Field
A non-empty set with addition and multiplication (may look different than the add. and mult. we know) that satisfies the following properties: 1. The set under addition is an abelian group with an additive identity of zero 2. The set minus the element 0 is an abelian group under multiplication 3. The set is distributive under addition and multiplication
Commutative Ring
A ring in which multiplication is commutative
(C, +) & (C, *)
abelian group
(Diagonal Matrices, *)
abelian group
(M(n,R), +)
abelian group
(Q, +) & (Q, *)
abelian group
(R+, +) and (R+, *)
abelian group
(R, +)
abelian group
(R-, +)
abelian group
(Z, +)
abelian group
(Q[x], +, *)
commutative polynomial ring
(R[x], +, *)
commutative polynomial ring
(Z[x], +, *)
commutative polynomial ring with unity
(Z, +, *)
commutative ring with identity
(Zn, +, *) modular operations
commutative ring with identity
(F, +, *) F for functions
commutative ring with unity
(C, +, *)
field, commutative ring with unity
(Q, +, *)
field, commutative ring with unity
(R, +, *)
field, commutative ring with unity
(Zn[x], +, *) modular operations
polynomial ring
(M(n,R), +, *)
ring with unity
