Definition:Ring with Unity
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Definition
A non-null ring $\left({R, +, \circ}\right)$ is a ring with unity iff the semigroup $\left({R, \circ}\right)$ has an identity element.
Such an identity element is known as a unity.
It follows that such a $\left({R, \circ}\right)$ is a monoid.
Also defined as
Some sources allow the null ring to be classified as a ring with unity.
Also known as
Other names for ring with unity are:
- Ring with a one
- 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.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970)... (previous)... (next): $\S 1.3$: Some special classes of rings
- A. G. Howson: A Handbook of Terms used in Algebra and Analysis (1972)... (previous)... (next): $\S 6$: Rings and fields
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 55 \ (2)$
- Paul Halmos and Steven Givant: Introduction to Boolean Algebras (2008)... (previous)... (next): $\S 1$