Summary of Properties for Groups, Rings, Fields

Summary of Properties for Groups, Rings, Fields

Under addition

1.  Closure

2.  Associativity

3.  Existence of an identity

4.  Existence of inverses

5.  Commutativity

Under Multiplication

6.  Closure

7.  Associativity

8.  Existence of an identity

9.  Existence of inverses (for all nonzero elements)

9’. No zero divisors

10.  Commutativity

Under addition and multiplication

11.  Distributivity

12.  0 ¹ 1

Algebraic Structure / Required property
Additive Group / 1-4
Additive Abelian Group / 1-5
Multiplicative (or just) Group / 6-9
Ring / 1-7, 11
Ring with unity / 1-8, 11
Commutative Ring / 1-7, 10, 11
Integral Domain / 1-8, 9’, 10-12
Field / 1-12

Algebraic Structure

Additive Group / Additive Abelian Group / Multiplicative (or just) Group
1 Closure / 1 Closure / 6. Closure
2. Associativity / 2. Associativity / 7. Associativity
3. Existence of an identity / 3. Existence of an identity / 8. Existence of an identity
4. Existence of inverses / 4. Existence of inverses / 9. Existence of inverses (for all nonzero elements)
5. Commutativity
Ring / Ring with unity / Commutative Ring
1 Closure / 1 Closure / 1 Closure
2. Associativity / 2. Associativity / 2. Associativity
3. Existence of an identity / 3. Existence of an identity / 3. Existence of an identity
4. Existence of inverses / 4. Existence of inverses / 4. Existence of inverses
5. Commutativity / 5. Commutativity / 5. Commutativity
6. Closure / 6. Closure / 6. Closure
7. Associativity / 7. Associativity / 7. Associativity
11. Distributivity / 8. Existence of an identity / 10. Commutativity
11. Distributivity / 11. Distributivity
Integral Domain / Field
1 Closure / 1 Closure
2. Associativity / 2. Associativity
3. Existence of an identity / 3. Existence of an identity
4. Existence of inverses / 4. Existence of inverses
5. Commutativity / 5. Commutativity
6. Closure / 6. Closure
7. Associativity / 7. Associativity
8. Existence of an identity / 8. Existence of an identity
9’. No zero divisors / 9. Existence of inverses (for all nonzero elements)
10. Commutativity / 10. Commutativity
11. Distributivity / 11. Distributivity
12. 0 ¹ 1 / 12. 0 ¹ 1