Product with Ring Negative

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 $x$.

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

Then:

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

We have:

$\ds \paren {x + \paren {-x} } \circ y$

$=$

$\ds 0_R \circ y$

Definition of Ring Zero

$\ds $

$=$

$\ds 0_R$

$\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.

$\blacksquare$

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 $6$