Ring is not Empty
Jump to navigation
Jump to search
Theorem
Proof
In a ring $\struct {R, +, \circ}$, $\struct {R, +}$ forms a group.
From Group is not Empty, the group $\struct {R, +}$ contains at least the identity, so cannot be empty.
So every ring $\struct {R, +, \circ}$ contains at least the identity for ring addition.
$\blacksquare$
Sources
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(c)}$