Product with Ring Negative/Corollary

Corollary to Product with Ring Negative

Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.

Then:

$\forall x \in R: \paren {-1_R} \circ x = -x$

Proof

\(\ds \paren {-1_R} \circ x\)

\(=\)

\(\ds -\paren {1_R \circ x}\)

Product with Ring Negative

\(\ds \)

\(=\)

\(\ds -x\)

Definition of Unity of Ring

$\blacksquare$

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $2 \ \text{(vi)}$