Product with Ring Negative
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Theorem
Let $\struct {R, +, \circ}$ be a ring.
Then:
- $\forall x, y \in \struct {R, +, \circ}: \paren {-x} \circ y = -\paren {x \circ y} = x \circ \paren {-y}$
where $\paren {-x}$ denotes the negative of $x$.
Corollary
Let $\struct {R, +, \circ}$ be a ring with unity $1_R$.
Then:
- $\forall x \in R: \paren {-1_R} \circ x = -x$
Proof
We have:
\(\ds \paren {x + \paren {-x} } \circ y\) | \(=\) | \(\ds 0_R \circ y\) | Definition of Ring Zero | |||||||||||
\(\ds \) | \(=\) | \(\ds 0_R\) | Ring Product with Zero | |||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \paren {x \circ y} + \paren {\paren {-x} \circ y}\) | \(=\) | \(\ds 0_R\) | Ring Axiom $\text D$: Distributivity of Product over Addition |
So from Group Axiom $\text G 3$: Existence of Inverse Element as applied to $\struct {R, +}$:
- $\paren {-x} \circ y = -\paren {x \circ y}$
The proof that $x \circ \paren {-y} = -\paren {x \circ y}$ follows identical lines.
$\blacksquare$
Also see
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 2$. Elementary Properties
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers: Theorem $20.9$
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Integral Domains: $\S 4$. Elementary Properties: Theorem $2 \ \text{(iv)}$
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $3$: Some special classes of rings: Lemma $1.2 \ \text{(ii)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 54.2$ The definition of a ring and its elementary consequences: $\text{(i)}$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: Exercise $4$
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$: Exercise $6$