Unit ring
From Wikipedia, the free encyclopedia
In mathematics, a unit ring or ring with a unit is a unital ring, i.e. a ring R with a (multiplicative) unit element, denoted by 1R or simply 1 if there is no risk of confusion.
(Some authors require a ring being unital by definition, and call a ring without unit sometimes pseudoring or Rng)
The integers Z and all fields (Q, R, C, finite fields Fq,...) are unit rings, and the set of all functions from a set I into a unit ring is once again a unit ring for pointwise multiplication.
Polynomials (with coefficients in a unit ring) and Schwartz distributions with compact support are unit rings for the convolution product.
Most spaces of (test) functions used in Analysis are rings without a unit (for pointwise multiplication), because these functions usually must decrease to 0 at infinity, so there cannot be a multiplicative unit (which must be equal to 1 everywhere).
Notice that a unit in ring theory is any invertible element (not only the unit element 1R). The term ring with a unit is nevertheless well-defined, because in order to define the notion of invertible, the ring must have a unit element 1R. Thus, a ring with "any" unit is always a unit ring.
