Ring Product with Zero
Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Then:
- $\forall x \in R: 0_R \circ x = 0_R = x \circ 0_R$
That is, the zero is a zero element for the ring product, thereby justifying its name.
Proof
Because $\struct {R, +, \circ}$ is a ring, $\struct {R, +}$ is a group.
Since $0_R$ is the identity in $\struct {R, +}$, we have $0_R + 0_R = 0_R$.
From the Cancellation Laws, all group elements are cancellable, so every element of $\struct {R, +}$ is cancellable for $+$.
Thus:
| \(\ds x \circ \paren {0_R + 0_R}\) | \(=\) | \(\ds x \circ 0_R\) | Definition of Ring Zero | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \circ 0_R} + \paren {x \circ 0_R}\) | \(=\) | \(\ds x \circ 0_R\) | Ring Axiom $\text D$: Distributivity of Product over Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {x \circ 0_R} + \paren {x \circ 0_R}\) | \(=\) | \(\ds \paren {x \circ 0_R} + 0_R\) | Definition of Ring Zero | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x \circ 0_R\) | \(=\) | \(\ds 0_R\) | Cancellation Laws |
Next:
| \(\ds \paren {0_R + 0_R} \circ x\) | \(=\) | \(\ds 0_R \circ x\) | Definition of Ring Zero | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {0_R \circ x} + \paren {0_R \circ x}\) | \(=\) | \(\ds 0_R \circ x\) | Ring Axiom $\text D$: Distributivity of Product over Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \paren {0_R \circ x} + \paren {0_R \circ x}\) | \(=\) | \(\ds 0_R + \paren {0_R \circ x}\) | Definition of Ring Zero | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0_R \circ x\) | \(=\) | \(\ds 0_R\) | Cancellation Laws |
$\blacksquare$
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 $2 \ \text{(iii)}$
- 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 $1.2 \ \text{(i)}$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 54.1$ The definition of a ring and its elementary consequences
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): zero: 2b.
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$: Exercise $6$