Unital ring
(Redirected from Ring with identity)
2010 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.
Nathan Jacobson, "Basic Algebra I" (2 ed) Dover (2012) ISBN 0486135225
Ring with identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ring_with_identity&oldid=39087
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Associative rings and algebras