Definition:Ring with Unity
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Definition
Let $\struct {R, +, \circ}$ be a non-null ring.
Then $\struct {R, +, \circ}$ is a ring with unity if and only if the multiplicative semigroup $\struct {R, \circ}$ has an identity element.
Such an identity element is known as a unity.
It follows that such a $\struct {R, \circ}$ is a monoid.
Examples
$2 \times 2$ Matrices
Let $S$ denote the set of square matrices of order $2$ whose entries are the set of real numbers.
Then $S$ forms a non-commutative ring with unity whose unity is the matrix $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$.
Also known as
Other names for ring with unity include:
- Ring with a one, or ring with $1$
- Ring with a multiplicative identity or just ring with identity
- Unitary ring
- Unital ring
- Unit ring
Some sources simply refer to a ring, taking the presence of the unity for granted.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, the term ring does not presuppose said presence.
Also see
- Results about rings with unity can be found here.
Sources
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $21$. Rings and Integral Domains
- 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
- 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: $(2)$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ring
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ring