Unital ring

2020 Mathematics Subject Classification: Primary: 16-XX [MSN][ZBL] unitary ring, ring with identity, ring-with-a-one

A ring with a multiplicative identity: an element $1$ such that $1x = x = x1$ for all elements $x$ of the ring. In many developments of the theory of rings, the existence of such an identity is taken as part of the definition of a ring. The term rng has been coined to denote rings in which the existence of an identity is not assumed. A unital ring homomorphism is a ring homomorphism between unital rings which respects the multiplicative identities. Any ring $R$ can be embedded in a ring $R^1$ with an identity by taking $R^1 = \mathbf{Z} \oplus R$ with multiplication $(m,r) \cdot (n,s) = (mn, ms + nr + rs)$ which has $(1,0)$ as a multiplicative identity.

A unital algebra $A$ over a field $K$ is an algebra over $K$ which is unital as a ring. As with rings, any $K$-algebra $A$ can be embedded in a unital $K$-algebra $A^1 = K \oplus A$.

A unital or unitary module is a module over a unital ring in which the identity element of the ring acts as the identity on the module.

References

• Nathan Jacobson, "Basic Algebra I" (2 ed) Dover (2012) ISBN 0486135225