Definition:Ring with Unity

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

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
• 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): $\S 1$