Unit ring
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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.
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[edit] Alternative definitions of a ring
Some authors (such as Herstein) require a ring to have a unit by definition. In those cases, a ring without unit is called a pseudoring or Rng.
[edit] Examples
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).
[edit] "Unit" versus "Ring with unit"
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 unital ring.
