Definition:Unit of Ring/Definition 1

Not to be confused with Definition:Unity of Ring.

Definition

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

An element $x \in R$ is a unit of $\struct {R, +, \circ}$ if and only if $x$ is invertible under $\circ$.

That is, a unit of $R$ is an element of $R$ which has an inverse.

$\exists y \in R: x \circ y = 1_R = y \circ x$

Also known as

Some sources use the term invertible element for unit of ring.

Also see

• Equivalence of Definitions of Unit of Ring

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains
• 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 6$: Rings and fields
• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55$. Special types of ring and ring elements: $(6)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): unit: 3.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): unit: 3.