Ring Exercise 1
We aim to determine whether the set of integers Z5 = {0, 1, 2, 3, 4}, equipped with addition and multiplication modulo 5, forms a ring:
$$ (Z_5 ,+_5, \cdot_5) $$
Since Z5 is a finite set, we can explicitly construct its addition table:
| a +5 b | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 | 0 |
| 2 | 2 | 3 | 4 | 0 | 1 |
| 3 | 3 | 4 | 0 | 1 | 2 |
| 4 | 4 | 0 | 1 | 2 | 3 |
And similarly, the multiplication table:
| a ·5 b | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 | 3 | 4 |
| 2 | 0 | 2 | 4 | 1 | 3 |
| 3 | 0 | 3 | 1 | 4 | 2 |
| 4 | 0 | 4 | 3 | 2 | 1 |
Both operations are closed on Z5, fulfilling the first requirement for a ring.
We now proceed to verify whether the remaining ring axioms are satisfied.
The First Operation: Addition (+5)
Modulo 5 addition on Z5 is commutative:
$$ a +_5 b = b +_5 a \quad \forall \ a, b \in Z_5 $$
It is also associative:
$$ (a +_5 b) +_5 c = a +_5 (b +_5 c) \quad \forall \ a, b, c \in Z_5 $$
The additive identity is 0:
$$ a +_5 0 = 0 +_5 a = a \quad \forall \ a \in Z_5 $$
Every element in Z5 has an additive inverse. For example:
0 + 0 = 0, 1 + 4 = 0, 2 + 3 = 0, and so on.
| a +5 b | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 |
| 1 | 1 | 2 | 3 | 4 | 0 |
| 2 | 2 | 3 | 4 | 0 | 1 |
| 3 | 3 | 4 | 0 | 1 | 2 |
| 4 | 4 | 0 | 1 | 2 | 3 |
Therefore, all the axioms for the additive structure of a ring are satisfied.
Note: The structure (Z5, +) forms a group since Z5 is non-empty, addition modulo 5 is closed and associative, has an identity element (0), and each element has an inverse. Furthermore, because addition is commutative, (Z5, +) is an abelian group, satisfying the first fundamental requirement for a ring.
The Second Operation: Multiplication (·5)
Modulo 5 multiplication on Z5 is associative:
$$ (a \cdot_5 b) \cdot_5 c = a \cdot_5 (b \cdot_5 c) \quad \forall \ a, b, c \in Z_5 $$
It also distributes over addition, both from the left and from the right:
$$ (a +_5 b) \cdot_5 c = a \cdot_5 c +_5 b \cdot_5 c \quad \forall \ a, b, c \in Z_5 $$
$$ a \cdot_5 (b +_5 c) = a \cdot_5 b +_5 a \cdot_5 c \quad \forall \ a, b, c \in Z_5 $$
Thus, all the ring axioms pertaining to multiplication are satisfied.
Conclusion
Since both operations satisfy the required axioms, we conclude that:
The algebraic structure (Z5, +, ·) is a ring.
And so on.
